Point out the health benefits, i.e. weight lose, improved muscle tone, improved cardiovascular health, etc. As for your parents/family and friends "worrying" about you point out to them that more motorists die on the nations roads then do cyclists/pedestrians.

As for the weather there is lightweight rain gear that you can buy to stay dry. If you add a pannier rack and bags you can pack the rain gear in one bag and just leave it there for when you needed. As for your "state" when you get to work. Check and see if they don't have a locker room somewhere that you can use. If they don't see if there is a gym nearby that you can join so that you can freshen up before going to work.

For going to school again, see if there is a gym on campus that you can freshen up before going to class.

Don't forget about securing your bike. Look and see if there is a good (preferably non-wheel bender) bike rack near where you work. If there isn't see if you can bring your bike indoors with you. If there is a good bike rack at or close to where you work and on campus close to your classroom(s) I would suggest that in order to save a little weight that you get two good U-bolt locks and leave them locked to the racks. Then carry a good quality cable lock to use to also secure the bike while at work and class as well as for your "shorter" trips.