I'll do my best with my limited knowledge - hopefully someone else can come and give more detail.
But basically, there is a range of radio frequencies that a scanner can pick up, which is the range that includes public safety, air traffic, etc etc. So yes, it will work nationwide, but...
No, different radio systems require different types of scanners. For example, if a fire department uses a trunked radio system, you need a scanner that can scan a trunked system. Basically, a trunked radio system is one that utilizes a small number of actual radio frequencies (which have to be assigned to an agency by the FCC), but allows the agency to have tons of "channels" that they can use, called talk groups.
Similarly for digital radio systems (which is what LAPD uses around where I am), you need a scanner that can handle digital transmissions (more expensive than most other scanners).
So try to figure out what type of system your local departments have (a lot of agencies post their radio frequencies online and actually encourage scanning), and get a scanner that can pick up that type of transmission.