This might be an unpopular opinion, but I'd start with Wrath of Khan (movie #2). Watch through Voyage Home (#4). Then watch #6. These are designed to be approachable. (Incidentally, the Stealing the Enterprise scene in #3 is one of my favorite sequences in all of Star Trek) :::
Then watch TNG -- you can start on season 3. You'll find more depth than you found in the films, both with characters and plot, at the cost of needing to pay closer attention.
If you like it, you can jump back to TOS (you'll already know the important characters from the films), or finish TNG, or try DS9. By that time, it'll feel more like just wanting new Star Trek to enjoy.
Stay away from movies #1 and #5 unless you already know you're a fan. There's value there, but it'll be hard to see unless those glasses are already rosey. spoiler What does god need with a starship? :::
Edit: I've been a lifelong lover of Trek, but as I think about it, this is probably how I first got into it as a much younger person. I don't know how many people actually start from TOS S01E01.