Well the missions haven't felt very satisfying and the enemies feel like bullet sponges and my character moves a bit slow… maybe these aspects are improved?
Mission design is improved, but only marginally. It will still feel like questing solo in an MMO from 2010. The main improvement is in the characters you interact with and their writing/dialogue, but I need to stress the STRONG caveat there that it's entirely dependent on whether the humour works for you. Some people love it, some absolutely detest it.
Bullet sponges - mostly yes imo, early on. But enemy variety improves too, so it can feel fresher, and like all ARPGs you're aiming for that endgame build where regular enemies start to melt and it's all about the boss grind.
Movement speed - I haven't played BL1 in such a long time, so never considered this and can't recall how it felt. I feel like it's not been a complaint I've ever had, but I played them in order. And I also love fast characters. There's an assassin class in BL2 that has a great movement ability to help him feel fast and reactive.
Ultimately I think if you're the kind of person that can enjoy ARPGs and the loot grind, then BL is a series worth checking out, but I'd jump in at the latest entry, or maaaaybe BL2. You're not missing out on the story. There really isn't much of one. Same way you wouldn't suggest playing Diablo from the first entry so you understand the story in 4.