Hi everyone here, I've had an argument with my English teacher at my school, so I want to ask you, which form of these sentences is better, more common, or if they have different meanings:

We tried to find/finding a cheap flight, but we were too late.

My brother tried to learn/learning Japanese at night school, but he gave up after two months.

The next thing I want to ask you is if it's more common to use "ATM" or "teller machine", than "cash machine"?

Thank you for your answers.