The only stuff that I missed out on was the DLC released for Fallout 3, and Borderlands 1 & 2. It's been on my to-do list, but I still haven't gotten around to it.
When you do get around to the series, I'd honestly go ahead and skip Borderlands 1 and just go straight to 2. Speaking as someone who played through both of them multiple times, the second does just about everything better. I know everyone was butthurt about it not being in the Handsome Collection, but I just see it as someone saving 15-20 hours of their life not playing an inferior product. The only connection between the two games is irrelevant anyways because of how non-existent the story was in the first. Also you don't have to suffer through one of the worst final bosses I've ever played.
Just some advice. If you really do wanna play it though make sure to get the DLC, the Dr. Ned set has more personality than the entire base game put together.