Ok I know people will say this is weird but watch through Star Trek Enterprise with Captain Archer since that was supposed to be a reboot for Star Trek that pulled in new fans. It explains the origin story of how humans got into space which I think helps a lot of newer watchers ground themselves in the universe easier. Make sure to make fun of it when it gets too absurd and always have a lighter nearby to hold up during the theme song.
Then go through TNG, Voyager and DS9 jumping between them every episode so you move through all three at the same time. I have been watching Star Trek with my gf this way and it is really fun to compare between the shows and it makes the bad episodes of any one particular series less exhausting because you get to take a break from that show.
Whenever you feel like it watch newer trek and catch up on it, I like all of it and some of it is really good.
Sprinkle in TOS episodes as you go, especially when there are direct callbacks to specific episodes (the SNW episode that echoes a TOS episode is a blast to watch right after watching the associated TOS episode they did such a good job).
I have been debating Enterprise, I really enjoyed the show. It was the first ST content that came out when I was an adult and the only series I watched every episode as it aired (save one or two that I missed because I got home late). My wife and I were just married when it released (less than two weeks) and she would sit on the couch and read while I watched it.