In a 5 gallon aquarium, you can have a single Betta. That's it as far as commonly available fish are concerned. All other fish require more space.
However, I do recommend a Betta as a first fish. They're fantastic little critters, easy to keep as far as fish go, and as mentioned, can be kept in a 5 gallon.
Do not get whatever fish you choose the same day as you get your aquarium.
Your aquarium needs to be properly cycled before any fish are added, and this process will take several weeks at least to complete.
Here's info on that subject:
The basics - http://www.firsttankguide.net/cycle.php
Best way - http://www.csupomona.edu/~jskoga/Aquariums/Ammonia.htm
Here are the basics you need to know for your Betta to live a long and happy life:
1. A cycled 5 gallon or larger aquarium is minimal for Betta health. Smaller tanks are impossible to safely and reliebly heat. You can cycle your tank with a Betta.
2. A filter is necessary in any tank in order to keep the water aerated and free of toxins. For Bettas it is best to set the filter flow to low, as they are not very strong swimmers. Another good option is a sponge filter. If your current filter is too strong, you can remedy it by covering the intake with a pair of pantyhose, putting a tall plant in front of the output, or follow these instructions to oufit your filter with a simple current/flow reducer: http://www.petfish.net/kb/entry/347/
3. A heater is essential. Bettas need warm steady temperatures of about 80F, to thrive. Constant fluctuation leads to stress, which inevitably leads to disease and often death. You can find an inexpensive heater for a small tank at any fish store.
4. In a 5-gallon or larger tank, weekly 25% water changes, using a gravel siphon are necessary. Water conditioner must be added to the new water before it is put in the tank to remove toxic chlorine, chloramine, metals, etc. Never change all of your Bettas water, as this shock your Betta, and ultimately weaken him. The Betta stays IN the aquarium during water changes.
5. Bettas enjoy a well planted aquarium, (use live or silk plants, plastic can tear their fins,) with a cave or two for exploring and resting. The tank should not, however, be stuffed with plants, as Bettas also need a good amount of room to swim. (And swim they will if given enough space.)
6. Feed your Betta a good staple pellet or flake diet, substituting frozen foods such as brine shrimp andand blood worms several times per week. Reserve one day per week for fasting.
When properly cared for, Bettas have been known to live for 10 years. I have personally known them to live for 6.
If you want fish other than a Betta, you'll need at least 10 gallons, though a 29 gallon is the smallest in which any sort of real community can be kept.
Here's a good list of fish that can be kept in 10 gallons, along with some stocking suggestions:
http://theaquariumwiki.com/Stocking_a_10_Gallon_Tank
Good general fish profiles:
http://www.peteducation.com/category.cfm?c=16+1911
http://www.aquariumlife.net/fishprof.asp