I just bought it myself, I've played BL2 but not TpS. I'd personally say jump right in and start with BL2 (playing Axton is probably best for solo but don't be afraid to just pick who you like).

I liked BL1 years ago, but didn't love it, I randomly bought BL2 for like £9 one day and it surprised me, it's (just imo) one of the best games I've played in recent years. I was having a gaming drought/burnout when I picked it up so I might be a little bit biased, but still I just fell in love with the game. IIRC I completed it at least 4 times solo and played tonnes of hours co-op, it was the only thing I played for a solid few months.

With BL2, I found the storyline to be so good, that it actually seemed to improve the storyline of the first game, as in characters from 1 make appearances and they are given more backstory and much more personality, and there's a few sad/powerful moments in the game that genuinely got to me. Not to mention, I love the BL style humour and Handsome Jack is a fucking amazing villain. This is a bit of a spoiler, but here's a snippet, I was fucking rolling because it's near the end of a game and he gives the huge dramatic speech, the game is full of these moments. (funny moment spoiler, not storyline spoiler)

While it may seem like a rip compared to just buying it on PC, it's a silly amount of content. BL2 lasted me literally months and the only DLC I had was Tiny Tina's (this is a fantastic DLC and I think it's generally seen as one of the best and most in-depth), basically you get two games with tonnes of content and all the (pretty substantial) DLCs included, as well as each game having 6 different characters. The TpS also has two fan favourites playable, Claptrap and Handsome Jack, and I think that's pretty cool.

I can't speak for TpS yet, but from what I understand it's more of the same and it's nearly as enjoyable as 2 then I'll be happy.