I'll back ResidentDante on this one and say that they all have pretty unique personalities because of the locales and soundtracks. Forza Horizon 1 also had some leftover cheesy "rival" characters that disappear in 2 and 3 -- they're kind of the over-the-top dumb personalities that you'd find in older Need For Speed games, and are meant to be sort of game progression goalposts. I think they're a lot of fun, but they are definitely stupid and appeal to the cheese-loving side of me, so your mileage may vary. I'm also a resident of the real-world Colorado, and seeing Horizon 1's fictional analog is entertaining -- they certainly nailed the general vibe and look of Colorado locales, but nothing specific is ever actually established.
Horizon 2 still has the most memorable licensed soundtrack in my own opinion (they're all great, though), but it's overall vibe lacks the same personality as the first game, IMO. Still, its Southern France setting is super gorgeous and stands out from the other two games.
Horizon 3 is probably the most robust and fine-tuned gameplay-wise, and its Hot Wheels expansion is insane and very fun to experience. If you HAVE TO go with just one for now, make sure it's this one. Fictional analog of Australia, personality is injected back in to be somewhere between the first and second games (just the right mass-appeal balance, I feel), and probably the most diverse locale yet -- forests, outback, a smaller version of a Sydney-esque city scape, all in one open world map.
For the Motorsport games? Just buy the latest one.