KH 1.5 and 2.5 consolidate all the games luckily so if you want to try playing through them all, well there you go! Well and then play 2.8 when it comes out, which is Dream Drop and a new episode of BBS that comes before KH3.

If you want to catch up by playing the games, the timeline order is

Birth by sleep
KH1
Chain of Memories
358/2
KH2
RE:Coded
Dream Drop
And then KH3

We don't know enough about KH3 to know if you'll need to play them all or not to understand the story. They'll probably make it easy to pick up since it's been so long since a big console release of a new KH. I believe they made KH 1.5, 2.5 and 2.8 so people can catch up though, so it may need it.

There are also lots of fan made youtube videos explaining it all too so if you really want to know but don't want to play the games, that's also an option.

I haven't played Birth yet myself so that part confuses me. I've generally spoiled it but yeah, I feel you! I wish they stayed all on one console but welp, here we are.