Don't stray from the one's with fees. They have the best rewards. There are plenty of sites that will calculate the amount you need to spend per month to make the fee pay for itself and many come with the first year free.
I've been putting absolutely everything on a CC since I was 18. I just pay the bill off every month. If you can afford the fee, I would look at the Chase Saphire, Starwoods Preferred, Southwest Card, or Capital One venture card. They fee ranges from 35-100 a year on those. If you are confident you can pay the bill, get one with a better rewards program and if you are worried that you need to make a purchase that maybe you can't pay off then use the one from the bank.
Having multiple cards improves the credit score as the line of credit is larger. Also, they most likely start you of with a low maximum due to your limited credit history. I still can't get behind opening up multiples and multiples as it just scares me but there is little to say that it ruins your score.
After I cash out my Capital One Venture for my honeymoon, I am going to switch to a the SPG and/or the Chase Sapphire only because my fiancee already has a southwest card.
Starwoods just merged with Marriot, so I'd stay away from that one.