One issue with BL co-op is that the loot is shared. I hated that, which put me off playing with randos. If you have a couple of friends, it's a very good experience, although you'll need good dedicateion, and a few hours to get through it properly. In BL3 co-op, the loot will be assigned to each players individually, much better IMO and stops loot rats.
BL1 is so long ago I can't remember much of it, BL2 is the refined borderlands experience IMO, and you can start with that. BL1 is OK, a bit 'slow'. Pre-sequel is good enough, and can be a laugh. The BL2 DLCs are mostly good too (Tiny Tina, Torgue, The Pirate one, ...).
The starting campaign is quite easy. True Vault Hunter Co-op is how it's suppose to be played IMO. It's casual enough, and challenging enough. UVHM can be frustrating, and will require farming for items, especially in OP levels.