What kind of bike should I be looking for and how much will I be spending?
Depends on the terrain, and how much you want to spend. You can get some very cheep bikes, but they are some what heavy and tend to clash gears like a orc mechanic. Some good brands are trek and giant. If your going by road, get a road bike, it will make your life a little easier. Or you could try for a very cheep bike second hand from a garage sale. As a general rule, anything over 20 years old will most likely work forever, and sell for peanuts. I have an really old tour bike that kicks ass! Also, try to get a bike suited to your gender, as their are male and female bikes, as well as unisex ones, but one for your own gender is best.
What other equipment will I need (helmet is obvious)?
Bottle of water, a way to carry bags, maybe a tire repair kit.
How long would 6 miles take to bike?
The general rule is figure out how long it takes for you to walk it, and divide by ten. Best way to take terrain and weather and such into account.
How difficult would a 6 mile bike ride be physically?
Easy, I have been doing it for years.
Legally, what do I need to know?
Bikes are road vehicles, so you are expected to ride on the road, not footpath, and stop at traffic lights. Although nobody enforces this, or aides by it, at least not here in Australia.