The general rule of thumb is one inch of fish per gallon. You want to go by the size of the fish when they are adults, not when you get them. Also, it depends on how clean of a fish they are, and whether they are bottom dwellers, top dwellers, or like to hang out in the middle of the water column. For instance, if you wanted to get neon tetras, they are a relatively clean fish, so you could probably put 10 - 15 neons in a ten gallon. However, if you wanted to put an algae eater (plecostomas) in there, you should probably only put 8 neons and an algae eater in a ten gallon.
Keep in mind that the more you do your water changes, the more fish you can keep in there. You want to do your water changes at least once a week, at least 20% of the water. The bigger the tank, the more stable your water quality should stay. I would go with a 16 gallon tank.
Also keep in mind that goldfish are VERY dirty fish, I wouldn't put more than 2 goldfish in a ten gallon tank. Even then, depending on the type of goldfish you get, goldfish can get very big and can outgrow a ten gallon very quickly.
Good luck, make sure you research the type of fish that you want before you get them, and research how to take care of a tank properly! A pet store that sells fish can generally educate you on the proper maintenance and set-up of a tank.
Have fun!