Jennifer,
It's ABSOLUTELY a good idea for you to start saving now, for two primary reasons: 1) it means you're forming a VERY good habit of controlling your spending and managing to save that will help you for the rest of your financial future; and 2) even a little money can add up over a long period of time! For example, saving just $25 a month from age 25 to age 65, growing at only 6%, accumulates to $50,000 of savings for your retirement! At 8.5%, it's over $100,000 for your retirement! However, it's important to note that if you have any existing credit card debt, your first goal should be to use the money to pay down that debt!
As for where to save - at this point, I think the best place to start is simply an interest-bearing savings or money market account. Get some practice just saving money in the account, and exercising the restraint not to use/spend it (unless it's a REAL emergency), and let it accumulate. Ultimately, you may decide that it's going to help you as a downpayment on your first home, or a deposit into your own dedicated retirement account in the future. Or you may have to tap it if you really need to - that's certainly better than taking on credit card debt at high interest rates! But for now, keep it liquid and available, and just get used to this practice for a lifetime of prudent spending and saving!
I hope that helps a little!