You are not alone as far as your weight being over a little. You can get many different opinions in this forum based on real experiences.

I think it's a good idea to start with mountain bike. Budget limit of $300.- maybe a bit tough considering you'll need a few other things like a helmet besides the bike. But it's still not a bad idea to go look at what bike shops have, so you can check the latest technologies. You may find a bike within your budget. If you have time to look around either thrift store or garage sale, you can sometimes find incredible deals, but you have to have minimal knowledge of separating a good bike from not so good ones. At least you want to avoid Mart bikes.

I see many new people posting their comments that they have re-discovered how fun it is to be on the saddle and peddaling again. You just have to experience it. Ask your friend or relative, if you can borrow their bike for a week or so. Many people even forget they have a bike in the garage.

The most important part is to get your butt on the saddle. The rest will follow.

By the way, wecome to the forum.