Many banks will give you a secured credit card, you usually have to give them a set amount, like $500 - $1000 and that becomes your "credit limit". You use your card for occasional purchases, pay it off before the due date, and after a reasonable period of time ( 6 months - 1 yr) you will have established enough credit that you should be able to get a mainstream credit card. Then you would cancel your secured credit card, and they will send you your deposit back.
If you are just trying to establish some sort of credit, you can do what I recommend to my clients ( I'm a mortgage broker so I have to help repair/establish credit all the time). If you have savings, put it into a locked in type of investment ( here in Canada we have G.I.C's: Guaranteed Investment Certificates) and then use the investment as security on a loan of the same amount. You can actually leave the "borrowed" money in the bank account and have the payments automatically withdrawn from it, then you don't even have to think about making the payments. The banks usually agree to this because they have the secured in a locked in investment that you can't touch, so they know that even if you default on the payment they will get their money back. Again, it usually takes between 6 mths to a year to establish the credit adequately, and all it has cost you is a little bit of interest ( meantime, you are still earning interest in the money you invested for the security!)
I use these techniques all the time, and it works well here in Canada. Hope this has been helpful.