There is an iOS app called Legal Heat that gives you the basic gun laws and carry requirements in all 50 states. It's updated as necessary. I'm sure there are plenty of others. And of course, there are lots of web sites with the same information.
Many states issue non-resident permits, but requirements and costs vary significantly. For example, a Utah non-resident permit is honored in 31 states. There probably is some combination of state non-resident permits that would cover almost all states, but there are some states that only honor permits issued to their own residents. Be very careful in those states.
Bottom line: know before you go. In states where you aren't covered by a valid permit, just keep the gun(s) secured in accordance with the federal transport laws until you leave that state.
Even in states where your permit is valid, there may be certain places where carrying your gun would still be prohibited, like schools and colleges, churches, bars, etc. Each state has its own rules about "prohibited places." For example, some states allow you to carry while in a restaurant where alcohol is served as long as you yourself are not drinking. Others do not. Some states are very strict about "concealed means concealed," and consider it a violation if your gun is even inadvertently revealed, such as by the outline being visible under your clothes. And federal law always prohibits guns in post offices, federal courts, and several other places.
Again, know before you go. It will take time and money to acquire the optimum number of state permits. Then you will need to plan your route carefully to know when you have to keep the gun locked up and when you don't.
A local gun club or shooting range would be a good place to start your research and planning. Here in New Hampshire, a local gun dealer regularly offers courses that qualify you for specific state non-resident permits. The Utah non-resident permit course is particularly popular.
Be careful out there...