This was written in 2016. If you read this in >= 2018 there is a good chance this is outdated.
Usually I don't post commercial stuff. But I think other people moving to the US might find this helpful. I did a quite comprehensive survey of the consumer credit card market and want to share my findings. If you live here for a year or longer you need to get a US credit card. Often enough this is the only form of payment accepted and international credit cards often enough fail for random reasons. I'm going to assume that you do not want the credit card for the credit, and that you will pay your total balance all at once and on time. If not, you should look into other offers. If not explicitly said, cards mentioned here do not come with annual membership fees.
Using the best offers I found, I'll get roughly $1000 cash back per year with the spending profile of a regular graduate student, so it is worth thinking for a few hours about a good strategy.
If you do not have a SSN
I do not know any online bank that allows applications without SSN, therefore you'll probably need to go to a brick and mortar bank. I have tried WellsFargo and Bank of America. WellsFargo only wanted to give me a credit card that has a fee (lol) so I went to Bank of America. They actually have a good offer: The BankAmericard Better Balance Rewards® Credit Card gives you $30 cash back each quarter. You can also go for the Cash Rewards Credit Card ($100 bonus for signing up online) that gives you1% on every purchase, 2% at grocery stores and wholesale clubs and 3% on gas for the first $2,500 in combined grocery/wholesale club/gas purchases each quarter. Do the math and you'll find that having both is the best option, but chances are high that you'll only be approved for one without a credit history. Many other international friends also went with Bank of America. They seem to be most welcoming to international students.
If you do have a SSN
With a SSN it is possible to get much better offers. Here are the best (and I did a LOT of research).
One card you should always have is the BankAmericard Better Balance Rewards® Credit Card, which simply gives you $30 cash back each quarter if you spent more than a certain amount (I think it was $30) per month. Of course, do not spend more than this limit, because you will not get extra cash back for that. A good strategy is to pay your phone bills with that (except you were smart and got a cheap mobile plain, which would be less than $30).
The best offer in the long term is Citi Double Cash which gives you 2% cash back on everything. No sign-up bonus. If you only want one card, and use it for a long time (several years), definitely go for this one. Therefore, the bar for other offers is >2%.
If you are into hunting bonus categories (e.g. amazon.com is often one category), you can get 5% in changing categories over the year.
There is the Discover It card and you will get 5% in changing bonus categories, 1% on everything else, $50 sign-up bonus and double the rewards in the first year. For the first year, this is the best you can get: 10% in bonus categories (e.g. Amazon.com, also applies for buying gift cards), and 2% on everything else.
Then there is Case Freedom, which also gives you 5% in changing bonus categories and 1% on everything else. Plus $150 sign-up bonus.
If you want to maximize your cash back, consider the Blue Cash Preferred from American Express as an additional card. Earn 6% Cash Back at U.S. supermarkets up to $6,000 per year in purchases (then 1%); 3% Cash Back at U.S. gas stations; 3% Cash Back at select U.S. department stores; and 1% Cash Back on other purchases. $75 annual fee. In my opinion, AmEx has the best web interface and this card is the most convenient to use and apply for. Getting the card makes sense, if you spend more than $1875 ($74/(0.06-0.02), the -0.02 is the cash back you would receive with another card) in supermarkets per year, which everyone probably does. $150 sign-up bonus.
If you spend a lot on traveling (airfare + hotels), like me, you might want to get a card that has extra high cash back in this category. I tried very hard, but could not find a good card that gives more than the 2% that I already get using the Citi card. The best I could find is the Costco Card, which offers 3% for travel expenses (4% on gas), but requires a Costco membership and you can only redeem your cash back by going to one of their stores, which was not an option for me.
If you spend a lot on gas, go for the Bank of America Cash Rewards Credit Card that gives you 3% on gas for the fist $2500. $100 online sign-up bonus.
Regarding Credit Card Acceptance
Visa and Master Card are ubiquitous. Discover > American Express. There are places that accept only Visa/Master Card, so you wanna have at least one of them in your pocket (e.g. the Citi double cash).