All planted aquariums need light, obviously. So the number one item on the list is a light. Lights come in all shapes and sizes and powers (see a guide to making your own cheap high powered lights). How much light you have on your aquarium will determine the rest of the list.
For low light aquariums (1.5-2 watts per gallon unless you have a very small or very large aquarium, then read this post on light in the aquarium) the equipment and supplies needed are minimal. Most are required for any aquarium:
- Light (at least 1.5 watts per gallon)
- Filter (biological and mechanical, the plants will take care of chemical)
- Substrate (can be plain gravel, though not ideal)
For medium light aquariums (2-3 watts per gallon) you will start to need to supplement your plants for them to grow healthy:
- Trace element fertilizer (such as Seachem Flourish or CSM+B)
- CO2 system or Flourish Excel
- Nitrogen, potassium, and phosphorous fertilizers.
It is important to remember, if you add fertilizers or CO2 to a low light aquarium in the proper amounts, you will still see an improvement in growth. There's no rule that says that low light aquariums shouldn't have CO2 or fertilizers. You just have to remember that plants aren't going to grow as quickly and won't need as much fertilization as plants in a high light tank.
As a beginner, I'd recommend starting out with low light and working your way up. Diving straight into a high light aquarium can lead to things quickly spiralling out of control. That, and it's fairly easy to upgrade your aquarium as you go.