Equipment

Keep in mind when you go to the store to purchase equipment that you want it to be fairly easy to clean and easily accessible. If it's not then maintenance may not get done on it. Lack of maintenance can and will result in equipment failure or breakage. Set a spending limit, it's easy to get caught up in buying things for a tank and it's not difficult to spend a whole paycheck. If necessary buy the things you absolutely need now, and the things you want at a later time. Now here's a list of equipment you will need:

A good filter. A lot of people prefer an undergravel filter, but I say that they are one of the easiest filters to forget about when it comes to cleaning. The best filter for the beginner is a hang on the back power filter such as those in the Penguin or Whisper line.
At least a 20-gallon tank; larger than 20-gallons if the fish you intend to buy require it, like an oscar. The larger the tank the more room there is for error and I can guarantee that you will make mistakes.
An air pump, especially if you pick a box filter that sits inside the tank.
A hood for the tank to prevent fish from jumping out.
Lights, fish and plants will need 12 to 14 hours of light per day. Fluorescent bulbs last longer than incandescent. They will only need changing once every six months, and they don't heat the water like incandescent, which makes controlling the temperature a lot easier.
A good heater. The water temperature for tropical fish will need to be kept at a constant 76 or 78 degrees. Coldwater fish, such as goldfish, like the water colder about 68 to 70 degrees so if you purchase coldwater fish then you may need to add a little fan to blow across the water surface during the summer months.
Thermometer. Those that adhere to the glass are easier to read, and there's no chance of them breaking.
Test kits for pH, ammonia, nitrites, and nitrates.
A vacuum for the gravel. Python makes a great vacuum that hooks up to your faucet, and sucks water and debris from your tank, sending it right down the drain.
A fish only bucket.
A safe level stand for your tank. If your stand is not level then you take the chance of your tank breaking once filled with water.
A net
And last of all a good water conditioner. I use Super Strength Tap Water Conditioner from Aquarium Pharmaceuticals, which removes chlorine, breaks the chloramine bond and detoxifies heavy metals. One drop per gallon makes the water instantly safe for fish.

Now for the fun part. Once you purchase the equipment you need then you get to buy the things you want. None of them are necessary and you can run a tank without them, but most people buy fish tanks for their beauty alone. VERY IMPORTANT: thoroughly run your hands over things you are considering and make sure none of them have sharp edges or are rough to the touch. If they are rough to you then there's a good chance your fish could scrape off a few scales. Here's a list of things you may want:

Gravel

Plants. You can choose live or plastic, in my opinion they both are beautiful. Just remember that live plants will require a little more care and that some fish like to eat them.

Decorations. Someone new to keeping fish should at all cost avoid using items that can be found in nature like rocks and seashells as you don't know what type of chemical or pollutant may be on them. Some items found in nature can also drastically alter the pH of your water causing all sorts of other problems.

If you purchased an air pump and are not using it on a filter, then a bubble wand is a beautiful addition.

There are a lot of other things out there that you can buy, but for right now until you are used to taking care of your tank I suggest that you stick to the basics. Why make it harder than it has to be?