Also I'd add a decent torque wrench onto the list of tools. The new barrel nut "should" be torqued. I say "should" cause some folks go by feel. I go the other route.
So you're looking at:
1. Armorer's wrench (I use a Magpul, great but $$$) to get the old barrel nut off, then new back on.
2. Small punch set (Roll Pin) to get the gas tube out of your gas block and later put back on
3. Allen wrenches OR another punch depending on how your current gas block is mounted, IE: set-screw VS pinned VS clamp-on
4. Bev Block (My preferred), Upper block, action rod etc. to hold your upper for disassembly and re-assembly. Again this is something some guys skip, but I have a thing against screwing chit up. Barrel nuts are torqued on and some are to the extreme end of the specs.
5. Decent bench vise, sturdily mounted
Depending on the FF forearm you choose possibly a manufacturer specific nut wrench that will fit the new barrel nut. Usually not very expensive, but they keep you from scratching up your nut. I've been buying SLR forearms lately so I'm still using the same wrench I bought with the 1st one. You won't need a set of gauges cause in theory the headspace was checked by the manufacturer. You'll end up getting a set later IF you don't head my warning below.
Be careful!! Cause probably the majority of AR builders started with something like a simple forearm swap. Then you've got these tools lying around and an extra forearm. The same tools for forearm swaps allow you to swap out barrels, so you're almost set for a complete upper build.............
and that's how it starts.