if the seat from your cousins is clean and has not been in a car crash it is fine to use the one they give you and get the other one for after.
i found that having the smaller one for while the baby was really small was easier because they sleep so much and it was easier to tote the seat and diaper bag around that way then to take her out of the seat and then grab the bags and stuff.

Check the car seats for "expiration" dates. I remember my older boys' having a five year limit. That was 10 years ago so things might have changed. The travel systems weren't available when I had my first (12 1/2 years ago) so we couldn't go that route. I'm so excited to be able to get one for our third due in Feb. I definetely suggest it!