Do private investigators get to travel and do field work?

In some situations for some assignments, yes, but it's generally work in he local area.

You never saw the Rockford Files? It's on reruns.

Do you get to travel? Only if the client is willing to pay for it and you have a reason to travel. Can you carry? Within the limits of the law,of course (in most states anyone can carry). Is it an office gig? Sure, if you consider the front seat of your car an office (most of the time, you are following someone and sitting around waiting to take incriminating pictures).

Are you prepared to invest money in a good camera with a long lens? You better be - that's how you get your evidence. Are you ready to get a license - in most states you need an investigator's license which may require on-the-job experience or some other qualification. For example, checkout the requirements to be a PI in Maryland - https://privateinvestigatoredu.org/maryland/

Do they 'get' to travel? Sometimes they have to, domestically, but it's not fun travel or even interesting. Most PI work is spying on cheating husbands and trying to find out if your employee is stealing from you. Not much else. It's not what you've seen on TV and in movies, dude.