There's a YouTube musician that I'd like to reach out to, to ask if I could purchase a license to use some of their music in my game, but I've never done this before and I was wondering if this is something I should get legal assistance with? Or is it as simple as sending a payment and getting written permission in an email? And I'd like to present myself as professional, so I am also wondering if there are perhaps industry or legal terms I should be aware of. I wasn't sure if this topic belongs here or in the Audio & Music forum, so I apologize if this is the wrong place to ask this.

A license is just a statement from the copyright holder allowing you to redistribute the work. So if you need a lawyer, it would be to make sure you understand the license given to you by the author.

Note: I'm not a lawyer, and this is not legal advice.

A lawyer is also important for understanding the implications of the clauses in the agreement, so you can negotiate for friendly terms.