Yes it will work across the U.S. as long as there is a Verizon tower within range. Like any cell phone, reception will be best in or near cities/towns, will get weaker the farther away you get from them, and have dead zones in remote areas. Compare the coverage map to your travel route to predict whether you'll be able to pick up reception in a location. If you buy an International Plan, it will also work in other countries.
You will have nationwide service on the Verizon NATIVE network. That means that there is no roaming at all and in places that have no Verizon native coverage, you will not have service ...or put in another way, you will have service only where you are receiving a signal off a tower that Verizon owns. So yes, you will have service everywhere in the U.S. where there is Verizon native coverage.
By going to the Verizon coverage map, you can see where the native coverage will be nationwide. If you check the "Prepaid" button and click on the "Refresh Map" button, the native voice coverage will be all the areas shaded in purple. The gray shaded areas and areas in white, you will not have coverage. Same if you choose the "Data Coverage."
However, remember that you will only have 3G data coverage and will not be able to access 4G LTE with this phone and Straight Talk service.
Verizon coverage map (copy-and-paste) ...go to middle of the page:
BTW, many RVers and OTR truckers use Verizon as it is considered to have the best coverage when traveling especially in rural areas.