You might need the stuff that jd mentions for the warm up and waiting for race to start, but once race is on you'll want to go light. I wear shorts (most go with longs) and a longsleeved base layer, plus a SS jersey on top. That's it. Nothing waterproof, you'll boil. MTB shoes are fine, just fit the same pedals as your MTB to your cross bike. I use eggbeaters and love their mud-shedding capability, but others use SPDs or Time ATACs.
TBH I don't use any waterproof stuff at all, just my regular cycling clothes, plus often the clothes I arrived in for the warm up/reccy laps, with race clothes underneath ready to go once the gridding starts.
Some of the running sections are steep muddy uphill - it helps if you have shoes that will take spikes. If you don't, go with what you have and change later if you start finding it a handicap.
Keep it cheap and simple for the first couple of races - after that you'll have plenty of ideas of your own what worked for you and what didn't. I pretty much guarantee you'll be thinking about tyres and wheels rather than clothes
