I'm pretty much obsessed with note-taking and to do list apps, although I don't know that it has made me anymore productive!
My favorite notetaking app is one you have to pay for - Notability - but I love it and use it all the time for all my work stuff. I'm trying hard to reduce all my paper notes and sticky notes. I also love the simplicity of the Notes app that comes standard on the iPad for making simple lists. It's so easy to use, I use it all the time, and that's where my massive books to read list currently lives!
I have also used Remember the Milk a little, but it's not one that has stuck with me.
I signed up for Springpad and downloaded Bamboo paper. Springpad seems like it could be fun to work with, but it was really slow!! Bamboo paper annoyed me because I couldn't type into it. I can see it being really cool for someone who likes to draw or doodle...and that someone is not me!! Although maybe I could use it to start exploring my artistic side.
I think it's always good to have a free notes app in your head when you are helping patrons new to the tablet world.
One thing I am noticing as I go through the things is how I hesitate to share and interconnect all of my apps and stuff with Facebook, Twitter, Tumblr, etc. It feels a little like oversharing to me - I'm not all that interesting, and I don't want everyone in my business! I wonder if it's a generational thing or just a personal preference.