Look for a recycled bike store in your area. Get fitted to a frame the right size for you in an aluminum frame road bike if you can find one that you can afford. If you are going to ride more than what you described, then you want a road bike. Financing a better bike with your parents may be worth the effort.
If you are going to really ride any miles, you want a dedicated road bike. You should invest in a good quality, aluminum-frame road bike if you want efficient transportation on pavement. I found that switching from a hybrid comfort bike to a low-end road bike increased my commuting speed 30% over my sixteen-mile daily ride.
I found a Trek 1500 that weighs about 20 pounds with narrow tires. I travel downhill at 42 MPH, level at 25 MPH, with an uphill speed of about 8 MPH. Total terrain average is 12 MPH.
I shopped around year-old models at 20% discount, but found a Trek 1500 demo model at a 40% discount in a 60 cm frame size. I am 6'2" at 195 pounds, a perfect fit. With accessories, I spent $800 with a tire repair kit, pump, and lock.
Within the first 1,000 miles, I had five flat tires. I replaced the tires with a Schwalbe Marathon Plus on the rear and a Continental Gator on the front. I had no flat tires over the second 1,000 miles. The tires cost about $100.
I spent more than $100 on Night Rider 15 watt headlight and rechargeable battery pack. I bought a back-up Cat Eye lamp with four rechargeable AA batteries. I also bought red flashers for the seat post and helmet, with a duct tape loop on the helmet to hold the LED lamp. Since I ride in below-zero windchills, I paid $160 for Hotronic electric footbed heaters in leather shoes that cost $100. I also wear layers of fleece with North Face waterproof pants and Spyder backpack cover, figure another $200.
So, I spent about $1,600 on everything that I needed for 2,000 miles of riding in the past year. Spread over five years, bicycle commuting will cost me less than $0.15 per mile, which is cheaper than driving a car and provides great fitness.