If they're worried about the cost of the cage, you'd better have them investigate veterinary costs before you actually get a guinea pig. Regular dog and cat vets won't do, as they get almost no training in guinea pig care. Exotic vets are few and far between, and expensive.
If a guinea pig is healthy, the cost of upkeep isn't too bad -- the cage set-up, hay, food and bedding. But if a pig gets sick, the cost of the care can run to hundreds of dollars and many hours of handfeeding a pig who's determined not to eat.
I'd suggest that you think about a pet that requires less money and care until you're old enough to pay for it on your own. Then, a guinea pig (or preferably two) will make a great pet for you.