Yes, opening a savings account would be a very smart decison. Placing money off to the side, especially in today's economy, should be very beneficial to you in the future.
I would suggest openinng and account with an online bank such as ING (ww.ing.com) or with any local bank or credit union that makes you feel comfortable. I suggest the online route because they pay higher intererst rates because their overheas s low.
I would also practice forced savings. By forced savings I mean enrolling in a direct deposit plan with your employer, they usually have give you the option of deducting a set amount or ercentage from your paycheck and directly deposit it into your savings acount. The nice thing about following a style like this is that you never really see the money in your checking account so you never really miss it or rely on it to be there to spend.
Finally, I would suggest setting up your online bill pay feature on your checking account if you have not done so already. The nice thing about this feature is that you can send a check or make an electronic tansfer deposit to your savings account. The nice thing about this is that it allows you to make small deposits / transfers when you save money. For example, if I save $1.23 on coupons ot through preferred buyer programs at the grcery store I send that amount to my savings account. Another example is illustrated through simpl budgeting, if I know I have $150 to spend on food for the week and only end up spending $133 I send the other $17 to my savings account. If you practice these little ticks you will be suprised how much money builds up in your account over the year!