Here's the low-down:
Goldfish have two basic varieties, the Fancy Goldfish such as Bubble Eyes, Pearlscales, Fantails and Ryukins, and the normal pond Goldfish such as Sarasa or Shubunkins. For a small starter set-up you want to be looking at the Fancy varieties, any good aquatics retailer will point you in that direction.
I'm going to be a bit of a stickler here and suggest that each Fancy Goldfish ideally requires about 35 litres of water per fish, purely because the little monsters create so much waste, so you want to aim for a tank that has that as a minimum volume. You can get away happily with two nice Fancy fish in there as long as you keep up your regular maintenance of the tank.
Goldfish have the potential to live 25 years plus if cared for correctly.
When you've set up your aquarium and had the filter running with a live bacteria additive such as Tetra Safestart for at least one week (ideally longer, up to 6 weeks) you can take a sample of the water (about 1/4 pint) to your local fish store and they'll test it for you, indicating whether or not the water is suitable to add fish to yet. If it isn't then depending on what's wrong with the water (high Nitrates, Nitrites, or Ammonia etc..) you'll either have to do a water change or add some more bacteria and taske another sample or water a week later. This should continue until the water is perfect for your first fish. Have a look here:
Fishkeeping - Fishless Tank Cycling and Avoiding New Tank Syndrome - General Guides - Articles
Provided your water is fine and you've selected a nice new companion and followed the fish store's guidelines on how to settle him into his new home, you'll want to watch his behaviour a few days and only feed a minimal amount every other day. Feed him enough that he'll eat within 2-3 minutes. With Fancy fish it's much better to feed a sinking granular type of food because flakes can upset their swimbladder when they uptake air from the water surface. This causes them to lose buoyancy.
At weekly intervals you'll be wanting to do a water change. 10-15% weekly should be more than enough and you might want to get your own water testing kit so you can see how your water is doing. Try to never remove more than 25% of your water in one go because the beneficial bacteria that helps to breakdown and process your fish's waste will be depleted. Never change all your gravel in one go for the same reason. Always add a dechlorinator to the water you add back to the tank. A good one is Tetra Aquasafe, and always top up your bacteria levels with the Tetra Safestart. I recommend Safestart because it has very little impact on the ecosystem in the tank, and you don't have to worry if you overdose it. When you clean the filter you want to remove a jug of water from your tank and rinse the sponges in their once a month or so. Don't rinse them under the tap because the chlorine in the water will agian deplete the levels of bacteria.
As for tank furniture, you'll be wanting things that aren't sharp that can tear your fish's fins. A nice hide and some silk plants would be nice. You can add real plants if you want but I hate them, lol.
A nice addition to the tank in the way of cleaning would be a snail, Apple or Trumpet or something as the fish won't munch on them like they're likely to do with a shrimp.
As for a companion for your fish it's best to stick to just other Fancy fish of the same size, provided the volume of the tank is enough for a couple. But if you must have something different you can look for something like a Butterfly Plec, provided they're kept in coldwater in the shop you purchase it from.
I hope that's given you some insight, lol, it's a lot to learn. If you want any clarification do give me a PM. I work for a large, well-known aquatics retailer and have a great passion for fish.
Happy fishkeeping