I'm getting started this year myself. Here's how I went about it.
1. If you were born later than '85 (I think that's the cutoff), you have to take a course- check the DWR website for availability.
2. A permit is not necessary if you're trapping fox, but for muskrat, it is. Get a tag anyway, you never know what's going to stumble into your trap.
3. Go to Sportsman's Warehouse- they've got quite a selection of traps and skinning gear, and it's ridiculously cheap (around $5-$7 a trap). You'll want to start with half a dozen #160 or #210 conibear (that's the measurement) leghold or break-neak traps. Muskrat, from what I hear, like inverted blind traps (check the proc- it explains it).
4. Get a book- either one at sportsman's, or order one from Amazon (usually cheaper).
5. Snowshoes are MANDATORY. You're screwed without them.
You can realistically get started trapping for somewhere in the neighborhood of $100, including tag. Be conservative, and shop around. Good luck!
p.s.- some tanneries will buy your hides if they're clean. Also, the state holds a furbearer tradeshow at the end of the season- kind of a modern day rendezvous. DON'T FORGET to register your traps with the DWR!