If I get a DUI in one state, will it show up in another?

Yes, an out-of-state DUI will show up in your home state. The reason is that typically all state motor vehicle departments have a network whereby each state can share information and communicate with every other state regarding matters such as revocation/suspension. Consequently, your out-of-state DUI will be treated as though in occurred in your home state.

You don't, however, indicate when you received these previous DUI's. The fact is that most states have what is known as a "look back" rule. Accordingly, if your previous DUI convictions happened outside of the stated timeframe, the court will not consider them for sentencing purposes, although they will remain on your permanent record. 

Since both civil and criminal charges attach to such a charge, you really should consult with an experienced DUI attorney in the area where you were arrested. Possibly they can get the charge dismissed or reduced.