Hello
i'm installing a new faucet in my bathroom.
1) Should I caulk unerneath the baseplate of the faucet? It doesn't have a rubber base, but a plastic one, so I was thinking I should caulk around it?
2) Should I caulk under the drain piece? like the circular donut you put in the drain, which screws in underneath.
1. I wouldn't bother caulking under the faucet. Water dripping under the faucet and onto the floor is much less of a problem than black mold growing on any silicone caulk you have there. If you do decide to caulk, buy the more expensive silicone caulks because they will have more mildewcide in them. Personally, I prefer Dow Corning 786 if it's available in your area. It comes in the only caulking tube I've ever come across that's truly resealable. However, to make it so you need to use a small hose clamp. If you buy any Dow Corning or Dap caulk in a resealable tube with a externally threaded cap on the nozzle, post again and I'll tell you how to reseal them.
2. Yes, It's a good idea to apply silicone caulk:
A) Around the sink hole under where the drain flange will be
B) On top of the rubber gasket that slides up the chrome tail piece to seal the bottom of the sink, and
C) All up and down that chrome tail piece that sticks down out of the sink, including on the threads near the top. The silicone will act like a lubricant allowing you to slide the rubber gasket up the chrome tail piece relatively easily. Without any lubrication between the rubber gasket and the chrome tail piece, it's like fighting with a bear to slide that rubber gasket into place. You don't need a lot of silicone on there, only a thin film to act as a lubricant.
So:
Start by caulking around the hole at the bottom of the sink.
Then, put something on an overturned pail under the sink so that you can drop the tail piece into the sink hole without the drain flange dropping onto that silicone you just put in the sink. Wearing latex gloves, slather up the tail piece (including the threads) with silicone caulk and drop it into the sink hole (but so the flange remains above the sink hole).
Now, apply silicone caulk to the top of the rubber gasket.
Now, grip the flange at the top of the tail piece and remove whatever's under the sink so that you can slide the rubber gasket onto the tail piece. Push the tailpiece down and the rubber gasket up until they're both tight against the sink. Now, put a pair of needle nose pliers into the tangs at the top of the tail piece and slip the large steel washer on from below and follow with the hex nut to hold the tail piece in place. Tighten the hex nut with a pair of slip joint pliers while preventing the tail piece from turning with the needle nose pliers.
Wipe up the blobs of silicone caulk, and then use paper towels to clean up the silicone film on the outside of the tail piece under the sink and on the top flange of the tail piece inside the sink. Paint thinner will dissolve silicone caulk even if it's partially cured, so finish up by wiping everything down with paper towels damp with paint thinner and allow time for the paint thinner to evaporate.
I'd leave the drain overnight at least before testing for leaks. Just put the plug in the drain, fill the sink and overflow channel and leave it that way for a while to see if anything leaks. Then pull the drian plug and drain the water into the pail you used earlier.
Throughout this post I refered to the thing you're installing as a "tail piece". That really should be called a "P. O. Plug".
Normally, bathrooms sinks use a combined drain and tail piece and the whole thing is called a P. O. Plug. Kitchen sinks use a separate strainer basket and tail piece.