Buy the largest aquarium you can reasonably afford. A small aquarium may be less of an initial investment, but it is much more subsceptible to water parameter changes. Things can go wrong very, very quickly in a small aquarium.
When "cleaning" aqaurium filter pads, sponges or biomedia be sure to use plain existing water from the aquarium. Using tap water to clean these items will kill off most of the beneficial bacteria. Under no circumstances should soaps or household cleaners be used in an aquarium.
Develop a regular water change schedule and STICK TO IT. The amount of water changed and the frequency of the water changes do have some flexibility, but make sure to be changing at least 25% of the water on a monthly basis.
Avoid over-feeding your fish. Any food that is placed in the aquarium should be eaten within 2 minutes at the most. Under normal circumstances, you should feed your fish no more than twice a day.
Make sure to practice due diligence with selecting new fish for your aquarium. Make sure the fish is compatible with existing species and will not grow too large for your aquarium.
Before making a fish purchase, observe all of the fish in the tank. If any of them are showing sign of sickness then pass on specimens in that tank.
Acclimate your fish to their new tank. You can float the bag in the aquarium for 30 minutes to allow the water temperature to normalize between the bag and the aquarium. It is also recommended to slowly introduce tank water to the bag to help with differences in water chemistry.
Use a quarantine tank for new fish whenever possible.
Read up on the 'cycling' process and understand how to tell where your aquarium is at in the process
Add stock to the aquarium at a slow, controlled pace. If too many fish are added within a short period the beneficial bacteria in the aquarium will not be able to keep up with the additional waste load.
NEVER overstock your aquarium. You will only be asking for trouble in the future. If are unsure of the stocking limits for your aquarium, please visit Fish Tank Planner to plan your aquarium stocking.
Do not use chemicals or treatments unless you fully understand both their immediate and long term effects.