I'd like to get into the Nanoha franchise, what would you suggest I watch?

Everything from the start. Season one is and forever will be my favorite entry in the series, and absolutely should be watched.
1. Mahou Shoujo Lyrical Nanoha
• Introduces Nanoha, Fate, some other recurring cast members.
2. Mahou Shoujo Lyrical Nanoha A's
• Introduces Hayate and the Knights.
At this point you can check out the films for an alternate take, but it's not required. You can watch these at any time after finishing A's really, doesn't matter much. These films are canonically movies created by the characters in-universe to explain the events of the first two seasons to other characters, complete with various embellishments and a dollop of artistic license:
3. Mahou Shoujo Lyrical Nanoha The Movie 1st
4. Mahou Shoujo Lyrical Nanoha The Movie 2nd A's
Back to the TV series:
5. Mahou Shoujo Lyrical Nanoha StrikerS
• Introduces... way too many people but most importantly, Vivio. First two cour season. I don't like this one a lot personally, but the next two series pull so much from it it may be worth watching for that just to understand who characters are.
The pseudo-spinoffs:
6. Mahou Shoujo Lyrical Nanoha ViVid
• Puts Vivio in the lead, changes the style of the show from sci-fi magical girls to martial arts tournament.
7. ViVid Strike!
• The one entry not to carry the "Mahou Shoujo Lyrical Nanoha" prefix. There are reasons for this, but it's too much detail to worry about now. New protagonist(s), Fuuka and Rinne. The cast of ViVid returns, for the most part. Sort of a ViVid sequel, but... it's complicated. It's also amazing, and possibly my favorite season since the original, or at least A's.
Upcoming:
8. Mahou Shoujo Lyrical Nanoha Reflection
• Upcoming film that takes place 3 months after A's ends. New material, not recaps like the previous two films.
9. Mahou Shoujo Lyrical Nanoha ViVid Season 2
• The continuation of Nanoha ViVid, which literally ended in the middle of a tournament. So this will help fill in some of the gasp between Nanoha ViVid s1 and ViVid Strike!. ...Probably.
If you reaaaaaaaaaalllly want to, you can hunt down the old OVA "Triangle Hearts ~Sweet Songs Forever~. It's a really crappy visual novel adaptation about bodyguards and hitmen and other utterly forgettable nonsense. The only relevance is that Nanoha herself actually originates with Triangle Hearts. Sort of. It's complicated. (A lot of things with Nanoha are complicated.)
The best thing I can say about it is that if you're familiar with it, it'll make some aspects of the first season (specifically, Nanoha's family) really amusing, but not in any way that matters in the long run. If you don't want to watch it (probably a good choice), you can check out this post instead, which gives a rundown: https://andrearitsu.com/2016/12/18/lyrical-retrospective-part-0-triangle-heart-and-lyrical-toy-box/
Overall the franchise is one of my top 10 of all time, so I very very very much recommend it. Just prepare for about 3 really intense genre shifts along the way.

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