I’ve also had to furnish on a budget (read, little to nothing) and it can be done.
As the others have said, check out garage sales (easy to do since you have a car), go to second hand and antique shops for bits and pieces. You can develop your own style.
Renting furniture is a waste of money. Fine if you needed something for six months and then were going to the other side of the country and didn’t want to be moving furniture, but otherwise it’s a waste of money.
Ask your girlfriend to help your choose furniture. It will give you something to do together, which can bring you closer together. She might also have home décor magazines which can help you choose what you want.
Have fun!