Like lead (yeah, I just wrote the lead page), copper is a rather inert metal. Indeed more so: copper is a noble metal. Noble though it may be, it still has a naughty side that likes to form ionic compounds, though it certainly gives some trouble getting there!
Copper has two valences, +1 and +2. Both form oxides, but only one forms a hydroxide stable at room temperature. The easiest way by far to make the +1, i.e., Cu2O, is by electrolysis of a copper anode in a chloride solution. I don't think any other anion can be used; nitrate is an oxidizer and likely destabilizes it, disproportionating to copper metal and cupric ions, while sulfate, being divalent, just vomits up Cu(I), disproportionating it and consuming the other half. My theory is chloride forms a soluble complex, presumably with the half-cell reaction Cu + e- + 4Cl- = CuCl43-, i.e., tetracuprochloride(I) complex ion, which, as it diffuses into the basic cathode region of the cell, precipitates CuOH, which being unstable autodehydrates to the yellow to orange Cu2O.

This is my latest batch, with some blue cupric hydroxide/carbonate forming in places due to corrosion (doesn't bother me, I'm just going to roast it in air). The lumpiness however is copper sponge that was plating through. I have no idea why, it was a neutral salt solution as far as I know. The liquor seperated from the Cu2O was green, indicating some sort of copper complex (possibly cuprous as mentioned above, which could've been produced if the solution was acidic). I noticed it was forming a blue crust, so I bubbled air through it for a day or two. This produced a blue precipitate and turned the remaining water a lighter shade of blue (not a chloro-complex, which is always green with this amount of salt in solution). I seperated the blue precipitate and boiled it in clean water: it dissolved to a blue solution before finally making some sense and breaking down to a black suspension, CuO (indicating the blue stuff was Cu(OH)2) in clear liquid. On drying I noticed it was a dark drab olive color, indicating the ever-constant presence of chloride.
I would presume cuprous oxide can also be made by neutralizing cuprous chloride, but since the latter is insoluble, it may take some time (although once it gets going, it should dissolve in the chloride solution). It's also difficult to get clean, dry Cu2Cl2 since it oxidizes to CuCl2 and, I presume, an oxychloride, when dried in air.
On to the second valence state. As mentioned, this can be produced by just about any other anion. I personally have used sodium sulfate to electrolyze copper, but the Cu(OH)2 is rather well deflocculated and won't settle. Running the cell near boiling might help, decomposing the hydroxide to black CuO soon after it forms. It would also improve conductivity by allowing more Na2SO4 in solution. Unfortunately it'll evaporate pretty quick too; a refluxing vent would help nicely. After some time, the surface of the anode becomes coated with Cu2O (which is conductive), indicating either metastable Cu2SO4 or a progression of oxidation: Cu(metal) > Cu+ > Cu++.
Cupric hydroxides can also be precipitated from acid solutions, such as nitrate and sulfate. Chloride forms a stable complex, see below. By diluting the copper chloride until it turns cyan/blue (forcing equilibrium towards copper tetrahydrate complex), blue copper hydroxide or carbonate should still be obtainable.
Cupric oxide, CuO, can be produced directly, but is slow and tiresome because it only forms in layers. Metal powder may suffice when heated in air to at least red hot. (Avoid reducing flames; copper is easily reduced by nascent H2 or CO in the flame!) Once the black layer is flaked off, an adherent layer of Cu2O is seen which can be used for a variety of semiconductor experiments, but interesting though it is, it's off the immediate topic. ;)

This is my jar of oxide. The color is, for the most part, black, but I can detect a slight hint of red to it, suggesting Cu2O or Cu content. There may also be Fe or Pb impurity (for mechanical reasons). After messing with it a bit, I don't think it's all that pure, maybe 90-95% CuO, with Cu2O and then Fe2O3 as the major impurities.

Copper oxychloride suspension, precipitated from CuCl2 solution with NaHCO3, and a *lot* of foaming.
Copper hydroxychloride, to be more accurate, is made by precipitating a chloro-complex copper solution. Copper chloride (CuCl2) alone is more like cupric tetracuprichloride, in solution. When hydroxyl is added, both ions precipitate more or less simultaneously, one as chloride, the other as hydroxide. The result is a crystal structure that has chloride and hydroxyl ions alternating, more or less substituting at will. (Around 19 Cu(OH,Cl)2 compounds are known, each with unique stoichiometry of OH:Cl.) This material pyrolyzes to a black powder as plain hydroxide or carbonate does, but it smokes from the freed copper chloride evaporating (it has a high vapor pressure, as many chlorides do). This can probably be leached out after pyrolysis, but that doesn't matter to you if you want oxychloride, which this paragraph happens to be about!
Copper tetraammonium complexes can be prepared in much the same way, as well as many other complexes (Paris green is a good example: hydrous copper [acetate complex] arsenate) and for many other transition metals.
Clearly, you can dissolve oxide or hydroxide in HCl to get chloride, but that's usually an extra step. However, you can't dissolve the metal in acid directly because it is noble, below hydrogen. Lead alone barely displaces it and needs to be boiled to get any appreciable reaction; copper doesn't go at all. You need to add something to kick it in the pants and make it bend over and cry... Oxygen is a pretty good way to go. You can simply bubble air (or if you really like it slow, just leave it sit open) and the oxygen dissolved will corrode the copper ever so slowly. A step up is hydrogen peroxide, which decomposes to water, providing oxygen right there in the solution. A more stinky source is a hypochlorite like Pool Shock (calcium hypochlorite 68% or so), bleach or TCCA (also sold for pool chlorination, trichlorotriisocyanuric acid if I spelled it all right!), but these decompose to chlorine gas on contact with acid so adding them to a solution, you have to add slowly with lots of stirring, and do it into the wind so you don't gas yourself WW1-style!
If you leave the solution sit with metal in it, the solution will turn brown (Cu(I)Cl4 complex I'm guessing) and the metal will be coated with a whitish substace - Cu2Cl2. This stuff is soluble in chloride solutions due to complexation, hence the change in color. After sufficient oxidation, the solution returns to clear deep green. Likewise, it can be reduced (given excess acidity) by more metal, kind of a re-proportionation to Cu(I). When excess water is added to a Cu(I) complex, it re-coordinates with water molecules and the Cu2Cl2 falls out of solution.
Copper chloride (or sulfate, for that matter) cannot be prepared by electrolysis of a copper anode in acid solution because the copper is removed from solution at the cathode, in preference to hydrogen. In fact, copper electroplating is performed with a strongly acidic solution. (In contrast, steel can be electroplated into acid with little, if any, metal plated back on the cathode, until much of the acid is used up.)

Here's some copper chloride hydrate, massive form made of acicular (flat needles) deliquescent crystals that appear to form a skin over the still-moist center. On recrystallizing this sample (which on drying and storage, finally turned bluish), I've produced a few hundred grams of clean (perhaps 99%) crystal as below, with about half remaining as a green solution that dries to a green to blue paste and when diluted strongly, hydrolyzes to an orange dispersion (FeCl3 > Fe(OH)3?). I've also managed to get copper sulfate crystals from it...
A small sample of some much nicer copper chloride I isolated. This was grown from a deep green solution as above, but with much more care, filtering often to remove the crop. Nasty acicular crystals, they take up so much volume (much of that being solution, by capillary action), so you squeeze the filter paper, crushing the crystals and squeezing out more than half the volume. Then you have to dry them, turning and breaking up clumps often because CuCl2 is rather soluble and a little moisture can cake up a lot of crystals. Drying is the most remarkable time because they turn from a moist green to this dry sky blue!

Here's a picture of one spiny crop produced from recrystallizing the first material. It's quite green and saturated with solution. The structure here is fuzz on all internal surfaces, plus a surface mat of crystals which fell in and more growth covered it over. Processing involves breaking up the mats with a stir rod and squeezing the stuff in filter paper. Although these crystals are about 1/2" long, they're very thin and few are over 1/8" length afterwards. The aspect ratio (thickness to length) gets better though.
I don't recommend preparing this from metal; acetic acid isn't real strong and I doubt its delicate organic structure much appreciates a massive oxidizer like hypochlorite! There are two common ways to prepare this: dissolving oxide or hydroxide with acid, and a double metathesis reaction. The first is obvious enough so I'll go right to the other. Dissolve calcium carbonate in acetic acid to form calcium acetate. Add slowly to a copper sulfate solution (CuSO4.5H2O is often sold as stump killer or possibly fungicide) to precipitate insoluble calcium sulfate, leaving copper acetate in solution. Filter and, if you want crystals, dry; if you want to cementate with lead, now is as good a time as any. This is the common process used for preparing lead acetate (to be bleached into PbO2 for preparing electrodes) from cheap chemicals.

Copper acetate solution; should be blue. Green color is likely a result of chloride contamination, as the solution was made from some of the above copper oxychloride, seen in the extreme upper-left corner of this photo. Update: it seems I have copper impurity in my lead oxide, for some reason. On refining a batch of lead acetate, I obtained a green solution which became blue as it became more concentrated. Blue isn't in iron's repitoire, so it was most likely copper. There wasn't much and it "crystallized" into a syrupy mush, so I didn't bother to persue it further (acetates seem to be lazy to organize into a regular lattice).

Copper sulfate crystals, evaporated from a solution made with sulfuric acid and CuO. Cyan to green color probably due to chloride contamination and partial dehydration. Crystals are hard, sharp and rhombic, be careful when digging through that acidic solution with bare hands! :o) Alternate methods: you can certainly use H2O2 with sulfuric acid. You can use calcium hypochlorite, with the advantage that calcium will form the insoluble sulfate immediately (whereas you have to add a specific amount of H2SO4 to precipitate it out of CuCl2 solution, if used), but it has the downside of contaminating with chloride, no good if you want a true blue product.

This is a large crystal conglomerate I grew, mostly hydrothermally. That is to say, I dissolved my stock of copper sulfate (which, as you can see, didn't look very nice) in water, got some nice seed crystals and started growing by evaporation. Well the hotplate I dry stuff on is cold on one side, so the crystals were forming on the cold side and dissolving on the hot side. I decided, sure, I'll add some extra crystals (which I collected during the first drying phase, at higher purity than anything I already had, being recrystallized) and keep on growing. A month or two later I decided I'd remove this thing, since it really is too big for the pickle jar I've been growing it in. Plus it's getting really raggedy with extra crystals, those punks. It should be kept in a baggie since it tends to give up and absorb moisture, turning crummy like the stuff above. I'm going to give it to my Dad to put in his rock collection.

Sodium cuprate solution. What's that, you didn't know copper was amphoteric? Why I'll be damned, neither did I. And it apparently is stronger than carbonate, because I was once attempting to produce copper carbonate directly, in a soda solution. It turned a wonderful blue, but instead of producing anodic corrosion it only plated copper sponge out! The above solution was made by dissolving black CuO in a concentrated NaOH solution. It's a nice cobalt blue, isn't it? A shade darker, "bluer" if you will, than the standard copper hydrate color. It used to be perfectly clear, but I guess it's hydrolyzed a little. It takes a lot of lye to hold the Cu in solution, so it can't be that strong an acid.