[OK: I see you've responded while I was writing this. That'll learn me for being so long-winded. First, here in North America I'm jealous you have access to A3 and the other A sizes. ;) Second, Epson Semi-gloss is at least as good as the Epson Luster I suggest. I actually prefer the Semi-gloss, but it's more expensive and less common here.]
I'm going to second and elaborate on the excellent advice previous responders have written:
Notice that the 3880 has two pure black ink cartridges. The matte black is specialized for use with matte surface papers. The photo black is for photo-finish papers -- anything with a bit of sheen or glossiness to the printable surface. At this point in time matte surface is probably primarily used for the reproduction of watercolour paintings, although a number of black-and-white photographers also favour it. More about matte printing below.
By far your safest course is to use Epson Ultra Premium Photo Paper Luster as your first paper. Your printer has a built-in profile for most Epson inkjet papers, including this one. This paper comes as close as any to minimizing the difference between what you see on your monitor and what you get on your print and closely resembles the commercial photo papers you've likely handled all your life. Once you get your workflow ironed out for this paper, and if you want to save a little money, you can buy whichever less-expensive clone of it is available in your part of the world.
Epson Ultra Premium Luster is an excellent example of a resin-coated, or RC, photo paper. It may look "plastic", or at least highly artificial, but it's actually just a standard wood pulp substrate with an ink-receptive coating on one side, followed by a polyethelene coating on both sides. Finally, a texture is stamped into the printable side to dispel glare. This makes for a tough, flexible, waterproof, and wide-gamut medium. If you class yourself as a documentary and/or snapshot photographer, rather than an art photographer, a good RC paper will likely be your stock-in-trade.
IAC, once you become proficient with Ultra Premium Luster and clones, you can begin to experiment. Among RC papers, words like Gloss, Luster, Semi-gloss (as Barbara mentioned), Satin, and Semi-matte simply refer to the artificial texture stamped into the printable surface. To use one of these non-Epson RC papers, simply tell the printer driver that you're using the Epson paper that most nearly resembles the non-Epson paper you're actually using. Hopefully, the paper's packaging or the manufacturer's web site will indicate exactly which Epson paper choice to make.
Leaving RC behind, non-RC papers, including matte surface and fibre photo papers, are both more expensive and more fragile -- these are definitely meant to be framed behind glass. In theory, these are also archival and are meant for the museum and fine art trade, but once you've handled a good example you may become hooked on the premium spread, so take care. ;)
Many non-RC but still photo black ink papers, including the high-quality baryta fibre papers many of us on this forum favour, will give acceptable, or nearly acceptable, results if you simply tell the printer driver that you're using Ultra Premium Luster, then set the platen gap to the appropriate thickness. Ilford Gold Fibre Silk is an example of a museum-grade paper that will work well using this approach.
As others have mentioned, letter-size sample packs are available from many companies. To use these you'll need to learn how to download, install, then access the specific profile for each paper from the manufacturer's web site. Ask here if the on-line instructions are not sufficiently clear.
The final frontier is to experiment with matte surface papers (which require you to switch to matte-black ink) with their limited gamut and less-than-black blacks. Be aware that every time you switch from "gloss" (AKA photo) paper to matte you're incurring a few dollar's hit in lost ink due to the printer flushing the black ink line.
[BTW: if you need to use your 3880 to print an occassional word-processing document on plain old letter paper, you don't need to switch to matte black. Just leave the paper type set to whatever paper it defaults to.]
Buy whatever sheet size you think you'll use most often. If you want to print small tests from time to time, using a razor knife and straight edge you can always cut a sheet in half or smaller, down to the minimum 4x6 inch (I believe) size the printer can work with. As you mention, cutting A2 in half will give you two A3s, cutting A3s in half will give you two A4s. Super A3 cut in half gives a non-standard, but quite usable size as well. That said, cutting without a dedicated paper cutter can be tedious, since you have to make the appropriate measurements for each sheet.
