For me, this was a trial and error kind of thing. I started my Freshman year off with an across the body satchel, but I very quickly figured out that my shoulder wasn't prepared to carry the weight of my text books, laptop, and everything else I had for that day all the way up the hill to my classes. I gave my dad the across the shoulder bag because he said he'd use it for work, and I grabbed a tote bag from my closet. This gave me the same problem. So, I went to Target looking for a backpack, and I found one for $7. It was a steal, but it only lasted me one year and my trip to Europe. So, this year I opted for another backpack.
I definitely think that a backpack is the best way to go, but that's my personal opinion. I might just be too wimpy that I don't have enough upper body strength to have a tote bag to go to class with. I dunno. I do know that my marketing book and pysch book are wicked heavy, and I'll need them on the same day without being able to go back to my room, drop one book off and get the other one. So, I'm going with a backpack. No question there. But a lot of people in college use a backpack. About an equal number have a shoulder bag or tote bag. It's just up to the individual. But there are definitely a lot of mature colors and prints for backpacks that you can buy for college students.
Let me know in the comments what you carry your books to class with!