What to bring with me when I start with a new company?

Enough credit or cash to get home from the farthest coast if you're put out on the curb for any reason.

 

I can't agree with this enough. Don't listen to truck stop bar talk. The company does not have to get you home, or to the state line, or whatever. They can fire you, or you quit, anyhere. If that's the case it's up to the (ex)driver to get a way home.
Another reason to pack light, nothing needs to be shipped home.
Now that said, one of the few good things about megas. They usually have a driver in the area you can ride with to get back to a terminal for a new trainer if it's just a trainer issue.

 

Man you said it right there. Personally I always have an irrational fear of nuclear war or natural disaster that would require me walking home to SC. Everything I have on truck will fit onto the waist mounted trailer I carry. Bivy, sleeping bag, water filters etc etc and most importantly enough water to get me across the mojave if I have to walk out of LA.