Caring for finches is relatively simple. They make wonderful birds for someone just starting out, yet are still very enjoyable for experienced bird owners. They each have their own personalities and funny little quirks! :-)
Finches don't require one-on-one attention like cockatiels and other larger birds, because they aren't tame. They're perfectly content socializing among themselves, and don't need or particularly want human interaction. Finches are naturally leery of people and may be a little skittish at first, although they do get used to seeing a familiar face, especially the person who cares for them most often. :-) I had a society finch that was comfortable enough that he would tug on my hair if I leaned close to the cage. He wouldn't sit on my finger, but he sure thought my hair would make good nesting materials! ha ha
As with all animals, finches need clean, fresh water available to them on a constant basis. I prefer to use cups to hold their water. Others use water bottles similar to those used in hamster or gerbil cages, but I've found them to be unreliable. Birds will use their water cups not only for drinking, but also to bathe in. Because of this, their water will get dirty (and pooped in) faster than it would if they just drank it. Be sure to change their water often, washing the cup before refilling it. A good rule of thumb is that if YOU wouldn't drink the water yourself, it needs to be changed. ;-)
Finches also require a constant supply of fresh food. I feed mine a nutritionally-complete blend of seeds, pellets, and eggfood. They also enjoy fresh foods such as dark green lettuce (not the light green kind, which is mostly water with few vitamins), finely-chopped hard-boiled egg (including the shell), apples, pears, etc. Spray millet is given occasionally as a treat.
Finches should be given a cuttlebone at all times. It provides calcium for nutritional purposes, and also for egg-laying. All healthy breeding-age females lay eggs even if she hasn't been with a male. The eggs are, of course, infertile, but she still needs calcium in order to prevent the eggs from becoming stuck inside her. Unless a stuck egg is expelled, egg-binding is fatal.
As far as their cage requirements, a suitable finch cage is longer than it is tall. Finches need horizontal space to fly and get exercise. They each need one square foot of floor space, so two finches would need two square feet. The spacing of the cage bars should be 1/2" or less. The cage should be kept away from drafts, out of direct sunlight, and away from "predators" such as cats. With zebra finches, it's ideal to have a cage that has a slide-in divider, or a spare cage on stand-by, in case they squabble (they can be somewhat territorial - more about this below ***). Provide several perches, but make sure the perches aren't directly over their food and water cups, or they'll get soiled. It's easier to keep the cage clean if you line the seed tray with paper towels or newspaper, and use a seed guard around the bottom of the cage to help keep seeds off the floor. Birds are, by nature, somewhat messy critters, but the mess can be kept to a minimum.
Please don't use chemicals such as household cleaners, aerosols such as air freshener and hair spray, or paints around your birds. Birds are much more sensitive to their environment than other pets are. This includes cigarette smoke and non-stick cookware, which can emit dangerous fumes. Vinegar and diluted bleach can be used safely as long as the area is well-ventilated.
If you want to breed your finches, here is a good website that has detailed information. Please note that I do not encourage frivolous breeding. Before considering breeding your birds, make sure you can provide a permanent home for the babies, if need be. Your 2 birds can quickly turn into 5-8 birds!
*** Zebra finches normally get along okay as long as they are kept in even numbers in groups of 6 or more, or in pairs. They also should have plenty of space and visual barriers in the cage. If you have an odd number of zebras, they will buddy up with a companion (regardless of gender), and the "leftover" bird will likely get picked on. They may also squabble over their territories, such as the highest perch, a particular nest, etc. If you're wanting to breed them, it's a good idea to have only one pair per cage, because two pairs may both want the same nest, or may see the other pair as a threat or competition. As mentioned, it's handy to have a dividable cage or a spare cage just in case they need a cooling-off period. Also, when adding new zebra finches to the same cage as your existing zebra finches, you'll need to remove all the existing birds from the cage. Then rearrange all the perches and other decorations so that the cage is completely different. This will "reset" the territories, and put all the birds back on the bottom rung of the "pecking order" ladder. Then put all the birds (new and existing) into the cage at the same time. You can't just add new zebras to a cage with old zebras, or it's pretty well guaranteed that there will be squabbles. This paragraph isn't meant to discourage anyone from adopting zebra finches, but to let you know ahead of time what MAY happen. They usually get along with very little fussing. If they do squabble excessively, simply move the "troublemaker" to its own cage. You may be able to reintroduce it at a later date, by following the steps above. Society finches, owl finches, spice finches, and shaft-tail finches are more passive and usually have no problems getting along with each other.
Disclaimer: I am not a veterinarian. Although I do have experience with birds, I am not "legally" qualified to offer specific medical advise. Please see your veterinarian for your pet care needs. :-)