Have you tried any student credit cards? Try sites like findcollegecards. There are plenty on a site like that you should be able to be approved for.

Try banks such as Discover, and Citi. You should be pretty good with them. Good luck!

Watch what you apply for because everytime a company checks out your credit score, it can actually hurt it. You should try to get a credit card with your bank. Since they are the keepers of your fortune they will usually approve with no history. It will be a small amount that gradually gets bigger, but it’s something to start with. If they don’t approve another option is to take out a small personal loan and pay it back over time, possibly putting the loan money in a savings account just to make sure you make the payments. Also, be careful if you get a card don’t max it out. Keep the amount you owe 1/3 of the available limit or less, because the closer the bill is to the limit, it can hurt your credit score.
I’m not positive, but I think the reason places run your credit when you are renting is to make sure you don’t have anything in collection agencies or a really bad score. I don’t know if it actually builds your credit.
As for time frame for things getting reported, it varies for different companies but I think the minimum is 30 days past due you get a warning and 60 days past due they report you.

your bank should be able to issue a credit card to you, even if it has a small limit it is better than nothing