I'm looking for a good bike to use to get around campus and use on the local trail which is about 10 miles paved and unpaved and goes around a lake. Preferably under $300.
If your campus is like most I have ever heard about, you want the cheapest bike that will do the job.
Bike theft is a common thing everywhere but campuses where a lot of young people are in need of bikes are certainly not less risky than average.
Most students I know look for second hand bikes, either at the university, online or where they grew up/have their relatives and friends network.
The brand of bike is less important, I guess you know better than I do what kind of bike you like for that.
I had a BMX bike for two years in college and no one wanted to steal it. To most common thieves it does not have a high value since it looks like a kids bike. They are fun to ride, can get you across campus through any terrain, and are small enough to fit inside closets and crevasess if you forgot your lock.
If that's not your thing. Try an older road bike off of craigslist. Something that you can maintain yourself, but not too shiny to get noticed. You want something light and with less moving parts so that way there is less to go wrong. Make sure you replace all the quick release parts.
I had a 20 year old Specialized road bike before going to the BMX. Before that, I had a 2 year old Mongoose that got stolen piece by piece.
Especially considering the theft issue raised by Willeke, I'd say hands down the bike for you is a rigid mtb. That's what I ride. I live in a high-crime city, and ride it on fairly technical trails.
I would buy used. You could probably get a decent used one for less than a hundred bucks.
I would probably put some skinnier 26x1.5 tires on it, depending on how rough the unpaved stuff is.