What can I do to protect my hands while felting?
You might consider a small piece of leather.  I've seen leather pot holders that would likely work, but any suitable size piece of suede of sufficient thickness would work.

You could also wear a heavy leather glove on the hand that you're holding the piece you're felting.

(note -- I don't do felting, but I've done plenty of other needlework projects ... including hand-sewing leather, so I know needles don't go easily into it unless it's that thin stuff they use for driving gloves and such)