For your first few bowls you'll probably go through a lot of 60 or 80 grit sandpaper to clear away the rough spots left by tearout. After that it's a simple progression to finer and finer grits, I usually go down to 400 or 600 then do a once over with some steel wool. From there I will either put on some sanding sealer or friction polish, and then a few coats of wipe on poly until it's as smooth as I want it to be.
For taking the tenon off the bottom you may want to build a donut chuck, there are plenty of examples around the internet, or if you are handy with a router a longworth chuck is another option.
I want to die quietly in my sleep like my grandfather, not screaming in terror like the passengers of his car.