Would this be a good shirt to wear under a vest?

I don't wear vests, but this top is lightweight enough that it would be perfect for layering. Just remember to order 1-2 sizes up as this top fits quite snugly.

is the material cheap and thin?

The material is thin. I wear them daily to work as I find it easier to wear this to cover my arm tattoos. They are comfortable to wear. I work in a hospital so it always a bit cold. I have been able to wear them under a vest all year long with not being too hot. I have washed them many times and they hold up well.

is it thick fabric or see through thin fabric?

It is thicker fabric. You cannot see through it.

I generally wear a 1X and want the shirt to be fitted but with breathing room; would I get the 1 or order up and get a 2X?

I would get the 1x. I did not have to get one size up like some of the reviews said to get. I got a 2x and it is not fitted on me. I do not like my t-shirts fitted. I normally wear a 2x.