I have no prior credit history. Does anyone know of a credit card company that doesn't require much (if any) deposit money down and has a high approval rate for first time credit card holders?
It used to be quite easy for students to get credit cards. When I was in college in the early 1980s, credit card companies set up tables in the student center and offered low-limit cards along with free t-shirts on an almost daily schedule. The Credit CARD Act of 2009 made this much more difficult for banks.
If you have a bank or credit union account, the first thing I would do is talk to them about getting a card. Some banks offer a VISA Student Card specifically for college students. My daughter was able to get one when she turned 18, just before starting college. The credit limit is very low ($200 for freshmen, increasing each year until the limit is $500 during senior year). After graduation, it converts to a regular VISA account with a limit that depends on post-graduation income and the now established 4 years of credit history. The VISA web site has a list of banks offering this type of card.
Now I will give you the same unsolicited advice I gave my daughter, and the same advice I think most others here would give you. For building credit, this kind of card is excellent, but you should still use it very sparingly, and pay off the balance every month. Make it a hard rule to never pay interest on a credit card bill. I told her to charge perhaps one or two purchases totaling no more than $25-$50 each month, and pay them off as soon as the statement arrives. This is much easier if you have a deposit account at the same bank, since you will be able to pay the bill instantly on line. Have your employer direct deposit your paychecks into that bank account, if at all possible.
Department store cards will appear on your credit report and is often much easier to get approved for. All my friends that have applied for a Macy's card have always been approved. If you are new to the country, department store cards are a great way to build history. Target and Nordstroms are two other department stores to look at. Target is my first suggestion since they carry every day items and will be easy to consistently put charges on the card to build credit history.
Store cards are incredibly generous with their approvals. I recently graduated and started working making around 100k and Bank of America only approved me for a $1000 credit limit... Macys however approved me for $2000 for everyday purchases and $5000 for large-ticket items like furniture sets. It may not sound so glamorous to have a specific store card but they all allow you to build credit, and a credit card really doesn't let you do anything a debit card can't unless you're buying things you don't currently have the money for.
Amazon is also issuing credit cards now, so with one of those there really isn't much you couldn't buy if you needed.