When my son went away to college, I gave him his first credit card, a Visa card with a $300 limit. It was technically my card, and I had listed him as an authorized user. He was allowed to buy clothes, books, and toiletries with his card. I paid the balance (and watched the bills to make sure he wasn't misusing the card). The card gave him some freedom, and I didn't have to keep sending him spending money. Once he graduated, we switched the card over to his name only.
You can get a pre-paid credit card for, say, $300 or maybe even $500, but you will have to keep very careful track of what you buy with it.
What are Prepaid Credit Cards?
I too wonder why you can't get a checking account.