How to Feed Your Bunny
Let's start with the basics. Your rabbit should generally have constant access to fresh hay, water, and dry rabbit pellets.
The average bunny eats about 1/2 a cup of dry rabbit pellets per 5lbs of body weight per day and will nibble at it throughout the day.
Fresh hay should make up the majority of your bunny's diet and should be available all of the time. Hay is also what your bunny will want to use as bedding.
Fresh, clean water should always be available. Period.
Fresh vegetables/fruits are also an important part of a healthy bunny's diet. Always exercise caution when feeding greens as some common foods can be poisonous, for example tomatoes and potatoes.
CAUTION when feeding greens: DO NOT suddenly introduce a huge amount of greens into a bunny's diet - they will get an upset tummy and the runs (which is especially difficult to deal with when you keep your bunny inside and not in a cage). Always use caution when feeding fresh fruits and vegetables, as excess greens can cause problems at any time.
How NOT To Feed Your Bunny
First of all, don't feed your bunny large amounts of any kind of greens (vegetables/fruits). One common mistake that many people make is feeding their bunny lots of lettuce. This is a bad idea. Large amounts of lettuce are bad for your bunny and will make it sick to it's stomach. Also, your rabbit will fill up on lettuce, which basically has no nutritional value to the bunny (which is bad).
The next thing you should know is to not feed your rabbit people food. Crackers and cake (and other people foods) are NOT ideal for your bunny, and can kill him or her.
A few of the more common vegetables and plants that are poisonous to your rabbit follows.
Bitter Cherry (seeds)
Tomato (Leaves and stems are toxic, but the actual fruit is okay in small amounts)
Walnut (green shell covering the nut is toxic)
(pictures borrowed from http://icanhas.cheezburger.com)