Just be careful and read the fine print with whatever credit card you do try and get.. A lot will give you a card but have a high annual fee or interest rate..

And to answer mortgage questions.. People under 620 score can get a mortgage but it's going to be a federal insured loan so you're going to pay a premium every month to get the mortgage with a low credit score.. Anything over 700 is pretty good to get into the lowest interest rates available.. Down payment really isn't a deciding factor to get a loan, just because you put half the value of the house down up front doesn't mean that you're going to be guaranteed a loan.. (it does help of course)...

I think the best option to try and build credit is to get a secured credit card though a reputable bank.. for one thing the credit limit never increases so if you're someone that tends to spend freely then this helps regulate that.. I have one for 500.00 that I used to build my credit over the last 2 years that I've lived in the U.S. and it really was only about 6 months after I had that card that I got my first car loan and then a house.. just make sure you use it and pay it off right away..