You're not missing much. I would honestly just stick with the order in which they were released, starting with Metal Gear Solid 1 on the ps1. It holds up well.

I personally disliked mgs2 a lot, but found 3 to be fantastic. 4 I also disliked for story which just draaaags, and 5 has the most fluid gameplay, which I did find fun, but the story was pretty meh on that one, and it ends pretty jarringly since they fired the creator mid production.

If you want just the highlights, play 1, 3, and 5.

The two PSP games are chronologically after mgs3, so I'd recommend playing that first to get the most out of them.

Mgs5 is basically a direct sequel to peacewalker on the PSP.