Thank you so much! This is the way I am going to follow. This was the easiest answer for me to understand. I'm sorry!
Most of Star Trek is episodic with little in the way of continuity between episodes so you can jump about a bit and sample the better offerings. My recommended viewing order is below but the easiest jumping on points if you're new to Star Trek are TOS, the TOS movies or TNG.
The 23rd century shows
The Original Series (TOS)
Set in the 23rd century aboard aboard the starship enterprise.
The Animated Series (TAS)
A continuation of TOS. It isn't always considered part of the official Star Trek continuity making it optional.
Movies 1-6 (The TOS Movies)
Set after TOS and TAS TV shows. "The Wrath of Khan" follows on from the events of the TOS episode "Space Seed".
The 24th Century Shows
The Next Generation (TNG)
Set in the 24th century about 70 years after the TOS Movies aboard a new starship enterprise with a new crew.
Movies 7-10 (The TNG Movies)
Set after TNG TV show. "First Contact" follows on from the events of the TNG episode "The Best of Both Worlds".
Deep Space Nine (DS9)
Set on a space station in the same era as TNG. Unlike other shows there is a lot of continuity between episodes making it difficult to jump into individual episodes. It's probably a good idea to watch some of TNG first.
Set on a ship that's lost on the other side of the galaxy. Set in the same ear as TNG but the show is mostly self contained due to its location.
The Prequel Show
Set in the 22nd century making it a prequel however I recommend watching it last.
The Reboot Movies
A reboot of the original series set in an alternate time line making them separate from the original and self contained.