Thank you, Ill start with two then.
Personally, I'd recommend taking the time and money to buy and do at least one playthrough of 1 first. It's a shorter game (which I appreciate, because it makes multiple playthroughs much easier, but it also means that doing a single playthrough of 1 shouldn't be that time consuming), and the story of 2 is a direct follow-up of 1's. Almost every single character from 1 makes a return in 2, so you'll be able to understand their significance better if you've played it. Plus there's a region from 1 that you actually go back to in 2, and you can't appreciate that without doing 1.
I started re-playing the Borderlands games when BL3 was officially announced; just finished my first playthrough of 2 & I'm getting ready to dive into TPS again. If I still have time before BL3 releases, I'll next be replaying them all three in second playthrough - which is when it gets really fun because you should, by then, have at least one skill tree maxed.