So where do you find out how much political favor you have? One of the missions unlocked a soldier base, but I need a certain amount of political favor to build.

Go to your TRADE tab, and in the lower window where you have "Minerals" and "Artifacts", there's another tab called "Special"... click on that and it will tell you how many Political Favour you have...

Thank you.

Now my next question is what missions do you do that let you gain political favor?

2 that i know of so far...
Dangerous Rescue (lvl 20+ i think)
and Priceless Transport (lvl 80+ i think) not sure really the pricelesss transport one is just a guess, im lvl 98 and i dont remember when it popped up

Also, the last mission of the rebels mission chain (that grants the anti-resistance base) also has a random chance of 2 political favor per attempt.