I have petellar tendonitis and my doctor said it would be ok if I go buy a bike. I do not plan on entering any competitions and I only expect to go out twice a week. I do not want to spend more than $300 as I am just getting into this.

If you're just riding on the street/paved bike trails, almost anything will do.  You can sometimes find good deals on used mountain bikes on craig's list, but if you buy new, you can probably get by with a Chinamart POS special unless you're riding off road.  If you get serious about it, you can get something nicer later.

I would look at Trek bicycles, I've got a 4300 that's been great for me.

Stretched out beach cruiser (wifeys)or a 65 Schwinn stingray 5 speed custom(mine)

Go to a good bike store (I would never buy a bike anywhere else) and you can get a good lower-level Raleigh or Trek hybrid in your price range.  I've seen Raleighs in the high $200's so it should fit your budget.  Hybrids are a cross between a mountain bike (too heavy for my tastes) and a road bike (too specialized for one thing...speed).  A hybrid is a good all-around bike for most people.  It's basically like a lighter mountain bike.  A good bike shop will fit the bike to you and even make adjustments like raising/lowering the handlebars, changing the stem or seat post or seat.  Don't underestimate how important these things are.  A good, well fitted bike will make a  HUGE difference in the enjoyment you get out of riding.  Every person who has ever ridden one of my Trek hybrids is just shocked at how nice they are and how good they feel to ride.