Printer resolution refers to how densely the ink droplets are laid down on the paper. The big numbers are more marketing hype than being really useful unfortunately. Once you get beyond about 1200 dpi, visual differences in the resulting print can be hard to see, although your printing times will increase dramatically. You can see a slight difference with glossy papers at about 2800, but beyond that, not so much. In addition, the really high resolutions typically don’t use the best photo-printing algorithms.
The choice of paper affects the final look of your print, as well as its life. Cheap papers, especially when used with dye-based inks, can shorten the life of a print considerably.
Paper surface is an important, yet personal choice. Coated papers (glossy, matte, pearl, semi-gloss and so forth) make colors vibrant and images look their sharpest. Uncoated fine-art papers give a mellower look, but many photographers prefer that look for displaying photos. If you aren’t sure what you like, buy a sampler pack of different surfaces and print out an image on them.
All printers are optimized for printing photos that are at a printing resolution between 200 and 300 ppi (printing resolution isn’t the same as printer resolution) at the size printed. You can set this when processing RAW. You generally have to change JPEG files (often at an unusable 72 ppi). Slight differences in prints from resolutions between 200 and 300 ppi can be hard to see without a magnifying glass, but outside of that range, differences can be obvious.
Also, size your photos properly for the print when using Photoshop and Photoshop Elements. These programs automatically resize the image to match what the printer needs, but they may not do it as optimally as you would. Lightroom and Aperture size your photos appropriately as needed when you print.
RAW files need more sharpening than JPEG files because they’re not sharpened in the camera, whereas JPEGs usually are. In addition, you may need to sharpen photos for glossy prints differently than for matte prints. Sharpening is a whole skill beyond a short article like this. Nik Software’s Sharpener Pro can simplify the sharpening process, plus its sharpening algorithms are very good.