Start off by picking a spot for your new aquarium, this will help you to decide how large you want to go. Keep in mind you need one gallon of water per inch of fish. With goldfish you should gave two gallons per inch.
After you have decided on the size, you should now be able to pick out the right size filter. I prefer the power filter over underground. You will also need a heater to heat water for tropical fish, unless you choose goldfish, which like cold water. You will need 5 watts per gallon, so if you have a 20 gallon tank, you will need a 200 watt heater. You will also need 1 1/2 pounds of gravel per gallon. Pick out decorations, plants, water conditioner, and bacteria starter.
Once you have everything you need, you can fill the tank. Place the gravel and decorations into the tank, take your time, you can be picky about how the inside looks. Use tap water to fill the tank about an inch from the top. Plug in your filter and heater and add your conditioner and bacteria starter. Make sure you set the heater at about 80 F. A theomometer will help keep track of this. Now that that is done, BE PATIENT. Your tank has to run for 24 hours before you can add fish.
After the 24 hours is up, go to your local fish store and pick up some "starter fish". These guys are hearty and will help your tank on the right path to sucess. More specifically, once you add fish, your ammonia will spike. The bacteria in your tank, which breaks down the ammonia, needs a chance to grow. By the starter fish emmitting wastes into your tank, the bacteria can grow. These usually will take about a week or two until you are ready for more, less hearty fish. Be careful, if you rush this process you will be risking fish death, cloudy water, and a frustrated fish parent. The more patients you have the better "luck" you will have.
After about a month you should have a very healthy thriving aquarium. Just keep in mind a monthly 25% water change will help keep everyone happy and healthy. You are on your way to a wonderful hobby!