Ok, here's some random trivia in no particular order:

You might already know this popular one, but Majel Barrett (Nurse Chapel, and the first officer in the pilot with Pike) is Gene Roddenberry's wife. She went on to play Troi's mother in TNG and is the voice for the ship's computer.

In season 3, see if you can spot when Spock suddenly wears something marketable and makes a point to talk about it, lol. It was controversial at the time and Nimoy was reportedly annoyed at first.

Also in S3, there is a "backdoor pilot" episode for a spinoff that never happened. They claim that it was not inspired by Doctor Who, but it seems suspiciously similar to me.

Also in S3, they were working on a tighter schedule and started getting more experimental with the camera work and direction. Some of it lands, some of it doesn't. You'll probably notice that some episodes feel different.

I don't remember which seasons the episodes are in, but Diane Muldaur plays two different roles in TOS. If you've seen TNG, be on the lookout for a young Dr. Pulaski.

Also, Star Trek was nearly cancelled early on due to the expense of it and concern that it wouldn't have broad enough appeal, but Lucille Ball herself saw the potential and said to keep it.