Speedy - I like the agreement you and your Dad worked out (should have thought of it myself, kid #1 is also a HS senior but we had my old car on hand and he drives his sister and sometimes the neighbor's son so it worked for us). Just to add to ZeroEnigma's advice:

For the short distances you mentioned to HS and the bookstore, basically anything will do. I'd look at hard-tail MTBs (heavier but more rugged, switch tires from knobbies to road tires) or a "performance" hybrid - the Trek is a good bike, although I'm partial to the Jamis Coda line. I second the advice to start with something good but relatively inexpensive; just like with cars, a lot of what you will learn with the 1st bike is what you like and don't like. If you have a friend or can find someone who knows something about bikes, checking yard sales and craigslist are options. Another reason to save some $$ is if you have college plans, not all schools allow freshmen to have cars, or the freshman parking is well away from the dorms, and you can't drive to classes anyway. Bikes are a lot better way to get around, but bike theft at many campuses is endemic, and bike parking may be limited to outside, two good reasons to go cheap vs. pricey. A good U-lock is essential, and try not to leave anything on the bike that can be removed easily.

To carry your stuff, look at messenger bags, backpacks with waist belts, or a crazy idea is a collapsible grocery bag pannier (designed to hold a standard paper grocery bag, also works to hold a bookbag or backpack and keeps the weight off your back).

You can probably wear street clothes, a wind or rain shell may be all you need in case you are caught having to ride home in bad weather. Platform pedals will let you wear street shoes/sneakers. I'd suggest ankle straps (with built in reflective tape) to keep pants out of the chain.

I'll also second the recommendations to have running lights - There are several economical LED/battery lights you can use for a headlight - I have a small Cateye Opticube on my helmet I keep in flash mode to aim at drivers (this is in addition to a BRIGHT Dinotte 200L - I am commuting in darkness and I need to see as well as be seen). For taillights, the $5-$10 LED blinkies at most stores, Performance Bikes, Nashbar are all good, although for about $20 ea, I really like the Planet Bike Superflash. I've strapped a blinkie to the back of my helmet with a zip tie; I usually run one light flashing (attention-getting) and one steady (easier for drivers to gauge your distance and speed).

Many bike shops and bike clubs hold classes on bike safety and bike repair (an excellent one is held locally via the school Adult Ed program and a local bike advocay group). Any good shop should be more than willing to show you the basics for repairing a flat and maintaining your bike (and if you get into it and have some aptitude, wrenching can be fun, save you some money, and provide part-time and summer employment).

Judging by the traffic around my son's high school and his comments, your biggest safety risk is going to be within a block of school from crazed parents running late (I drive past the school to get to my job and see examples almost daily) and your female classmates paying more attention to their cell phones than to their driving. I usually don't recommend riding on the sidewalks, but depending on your streets, they may be safer than sharing the road. Pay VERY close attention at driveways and intersections - drivers and pedestrians aren't expecting a fast-moving cyclist and will not be looking for you. Assume you are invisible and ride accordingly (reading traffic is also a skill that will help when you are driving).

Good luck, hope it works out.