What's the proper way to get a coin graded?

I would start with a coin shop. Tell them you want to know the value. This will give you a rough idea of the value. If it is worthwhile then you can get it graded at one of the services.

There are several on the internet to choose from.

take it to at least two reputable coin dealers, and then cross check the rating against standards on a numismatic web site.

Is a quick “value check” (not an in depth evaluation) typically free?

That depends on the dealer. If it is someone you deal with regularly, I am sure they will give you a ballpark figure to keep you as a customer.

Send it in to the USCG

USCG is not a recognised third party grader. They grade and buy and sell their own coins.

Check this report.

coin graders

Grading and packaging can be expensive, you need to get quick estimate from a couple of coin dealer.