Watch advice

I should state I don’t know a ton about watches.

This was my great-grandfather’s watch, his initials are on back. It appears to be a Vintage Hamilton Tonneau wristwatch. I don’t think it’s very valuable but would like to get it working again and such.

I guess my main question is do I get it cleaned? Freshen the old looking dial? Is that a thing? What steps would one recommend to bring back to life but maintain its vintage/family feel. Thanks y’all