HI there
I strongly suggest you to get a state ID as soon as possible. You just need to go to your closest DMV and show whatever papers they require in your state (requirements vary by state; you can know easily what they want by browsing their website). It's a rather painless process, except for the wait. Then you'll have an ID that is exactly like the ones American have, except it will be written in small prints "temporary visa holder". Once you have it you don't need to have your passport with you for anything. I take the plane within the US on the basis of this state ID. It will make your life much easier because people won't look at you funny anywhere since they'll assume you're a US resident and not a clueless foreigner (even though, as I said, it is written in small prints that you're a temporary visa holder.)
Also as others have suggested it's just very simple to ask for the driving license at the same time if you already have a license from your home country. In my state I needed to take a twenty minute test where they basically ask me whether you should stop at red lights.
In most states you probably won't have enough documents to ask for the state ID in the first weeks you arrive, but as soon as you'll be here for more than a month you'll be fine (depending on states, you may need to show a variety of cards and bills proving your identity and residence, which you usually get after a month or so.) It sounds intimidating at first sight, but it will make your life much easier!