(I have suggested specific brands by name in accordance with my personal experience. Other options are available; ask other hobbyists (not sales people) before you buy.) Remember: DO NOT BUY ANY FISH until your tank has been set up and cycled for a few weeks. Use this time to plan which fish you will buy and to determine their needs.

Of course you will need a fish tank. Get the largest you can afford. You will also need a cover and light. If you are not putting live plants in, the cover and light that sometimes comes with the aquarium is fine. If you want live plants consider getting a clear lid and two strip lights (one flora tube for the back and one daylight tube for the front).

A stand. Use only stands made especially for fish tanks. Do not use other furniture because water and tanks are heavy.

Heater (5 watts per gallon)
Thermometers (get more than one)
Mag-Float (algae scrubber)
Nets (the right size for your fish)
Siphon gravel vacuum
New unused buckets
Feeding ring (optional)
Books
Fish food (I recommend Omega One Freshwater Flakes, Hikari wafers and TetraMin)
Kosher salt, no additives (depending on fishes’ needs)

Water Quality Tests:
Quick Dip test strips (not as accurate as liquid tests, but easier)
Ammonia Test Kit
Nitrite Test Kit
if not using test strips, you will also need
Nitrate Test Kit
pH Test Kit
Hardness tests (GH and KH)

Other Chemicals:
Stress Coat water conditioner (by Aquarium Pharmaceuticals)
A.C.T. by Mardel or Nite Out II or other beneficial bacteria solution (optional)

Filtration:
Aquaclear/Fluval (hang on the back) Filter; (if in doubt, go a size larger)
Aquaclear /Fluval foam insert
Aquaclear /Fluval BioMax insert
Aquaclear /Fluval carbon insert
OR buy bulk charcoal and bulk ammonia chips
If you decide to go with a canister filter, I had great success with a Filstar XP from API. It is far better to chose a filter one size too large than to purchase one a little to small. Go big.  Purchase media accordingly: carbon insert (chemical filtration), foam insert (mechanical filtration), “BioMax” or other ceramic media (Biological filtration). You will want a felt “water polishing” pad if you have cloudy water to clear.

For Air Bubbles (optional):
Profile brand air pump (size depends on how many stones you will be running, not the size of your tank)
Airline tubing
Blue (glass) air stones (round or rod styles)
Air control kit (valves)
2 way air control valve (optional)
Check valve
Battery operated backup air pump (optional)

 

Decorations:

must specifically state “safe for aquariums”.
Plants and Caves for cover and housing (no wood collected from the wild, unless boiled and certain to be free of toxins)
Glass marbles, quartz or slate rocks are o.k.
Gravel (no sand), deep enough to hold plants down.

MEDICINES I RECOMMEND, as needed:

for parasites:
Aquarisol (ick preventative)* EDIT: The company that makes a competing product bought out the Aquarisol company and quickly discontinued making it so you can’t buy it anymore. Try CopperSafe instead.
Maricide
(although commonly recommended, I avoid Clout and Jungle Parasite Clear, because they are very strong and can kill your sick fish)

for bacterial infections:
Tri Sulfa
Marycin2
Melafix
AntiBacteria by A.P.

for true fungal infections:
MarOxy

first aid charts and disease guides (Mardel puts one in with medicines. It is very helpful.)