Agreed that Ubuntu is probably the most newbie-friendly. If you want to be familiar with *nixes in general, first off, spend as much time on the command line as possible. To get a feel for the Redhat side of the world, try CentOS. I'd avoid Fedora - it's known as "bleeding edge" and you never know what will break.

Also consider FreeBSD (just don't call it Linux). It does take getting used to, but one of the things I really like about it is that it has a stable core (that is now much easier to update), and source-based add-on software (ports), so you generally are pretty up-to-date in that area. The downside is that Flash can be tricky to get working.

And if learning all about Linux is your goal, set up a machine using Linux From Scratch. It's just a learning exercise, though. Don't expect to keep such a machine for day-to-day use.