Ditto on hitting up your bank. If you are otherwise in a financial position to buy a house, you must have a nice chunk of money saved. Go apply for one of their regular cards. If you use chase, ask for a Freedom Visa. If they reject the application, talk to your banker and ask him/her to put in a special request. This request will include information about your bank accounts and how long you have had them, which is information that is not considered in the original decision. If doing a special request, go for a small credit line ($500.00). If you use the card as much as you can, you will get a CLI very quickly.
This is how I did it. I was 24 when I got my first card (a Chase Freedom Visa
) with an income of 25K per year (I was a graduate student). It took some doing back in 2009 with the financial crisis still going on. Right now (2 years, 3 month later), my credit score is 735. The median is 720. So there you go, it takes about two years to reach median credit, if you do everything right.
Also, apply for a second card from someone else after about 6 months with your first card. I like Discover More myself. It is a bad idea to let one lender control the entirety of your credit history. If you do, they know that you aren't in a position to cancel the card without forfeiting a BIG chuck of your credit score. Also, this is a reason not to go for a card that you don't want as your first.