How do you say "do/am" in Mandarin?
I'm trying to say things like "I am an American." or "What do you do?"

Question

Answer

In general, the verb "to be" is 是 whereas the verb "to do" is 做.
Specifically, your examples can be translated as 我是美國人 and 你是做什麽工作的, respectively.

Answer

I am an American. 我是美国人。
What do you do？ 你是做什么(工作)的？

Answer

I am an American = 我  是  美国人。 = wǒ  shì  měi-guó-rén.

What do you do? (What's your job) = 你  做  什么  工作? = nǐ  zuò  shén-me  gōng-zuò?

Answer

Totally agree with above answers. Just one more supplement.
What do you do?    你做什么工作？  will be more concise to your question.