Some banks- or most do offer a secured credit card. It means-that you deposit your own money usually $250-$300 and spend it. After, you get a monthly bill with a minimum monthly payment that must be satisfied on time.
Building credit -the number one key to a successful credit history is to never have delinquent payments on your history. Also, when you are applying for credit cards- be very selective in your choice. The best credit cards are the ones that have zero interest for at least 12-15 months-this will allow you to spend the credit money, pay it back with no additional interest accrued on it. Also, I like the credit card that give me either cash back , rewards, or miles- it all depends what kind of a spender are you and where it makes sense for your daily saving. I enjoy the cash back rewards- I get a percentage cash back after the spending I do.
However, I recommend that you try to budget your lifestyle and try not to use credit cards often as you don’t want o accrue debt with high interest rate or a high variable interest rate. Use it wisely- pay it back on time- pay it more if you can-don’t have more then 2-3 credit cards especially if you are just building credit history. The worst credit cards from my own experience when I was in college are the store cards- they all sound very good especially when you want to save -it their interest rate is way too high-those are the cards that get you into too much trouble.
Best of luck