Hi there.

The first thing that will help you a bit is to take a look at the Compendium, specifically the Challenges section. That provides some basic info and lists them all. (For instance, challenges only show up M/W/F.)

As for re-rolling, that can be done when you have an active challenge by clicking the button just underneath the challenge that's labeled "reroll". It will replace that challenge with a randomly selected one.

Finally, as far as seeing your own progress, there's not a good in-game solution for this at the moment. As a temporary solution, I wrote a quick web app that lets you use a log file that the game generates to list which challenges you've done. Note that if you've disabled crash logs in the options menu, you'll need to re-enable them and restart the game in order for a log file to be generated.

Hope that helps, and welcome to the game!