I haven't watched the TV series, but I'm a big enthusiast of the games.

I'd say start at the start, especially if you're alright with more old-school games. The first two are (sometimes punishingly hard) isometric RPGs that I personally had a blast playing. Some of the later games have little references to them that you might miss, too.

Get Fallout 2 on GOG instead of Steam if you can, the version on Steam is the censored European version that removes all the child characters and quite a bit of content with them (since you had the option to kill them in the game's open world environment, despite the game heavily penalising child murder).

If turn-based combat and 90s graphics really repulse you (totally fair, I know they're not everyone's cup of tea), you could try playing the 3D games instead, which are essentially first-person RPGs with FPS elements. Fallout 3 is alright, the story isn't that good, and the gunplay just feels a bit off, but the environments are really well-crafted and are fun to traverse.

If you have to play one Fallout game though, I would pick Fallout New Vegas. It's mostly made by the guys who made the first two games, and they really really knocked the ball out of the park. It's still got the slightly awkward gunplay of the third game, but everything else is really well done. The story especially is probably one of the best video game narratives I've experienced (putting it up there with those of Disco Elysium and Undertale).

I haven't played Fallout 4, so I can't really give you any recommendation there.

The games are pretty dated

There is a good chance you won’t enjoy them, see if it’s something you’d like first and pick the one that looks most fun