Well... I'm the only immigration law paralegal in town and there is kind of a crazy saying here about me - the best immigration lawyer in the area isn't a lawyer at all. My supervising attorney has let me run with it and I've had a blast doing it.
What do I enjoy about it? The complexity, the need for absolute precision in preparing a Petition or Application submission package, playing the game with CIS. Learning. Learning. Learning. And did I mention learning? You'll never learn it all. It is too complex.
When I joined the firm, the attorney here thought immigration law was like social security law - paper only. Not flipping hardly. It is far closer to tax law - and for many of the same reasons.
What do I dislike? Having to pull information out of clients. Second would be having to turn people away because they can't afford the legal fees - especially when I can see the fear in their eyes. That's tough for me.
Yes it is very challenging. As to advancing, start off with a firm that does business immigration and employment related visas (especially super high tech) - those advance.
I do primarily family based, E-series (business/investment), R-series (religious), and the daily things (green card renewals, applications for citizenship, etc.) Ah - most satisfying - when a client comes in with his Naturalization Certificate.
Hope this helps.