Just re-read your question.....
For overall basic shaping; mostly rasps, spokeshave or drawknife, a bandsaw is handy too

Logier or Ariou hand stitched rasps are the best.
Barrel channels can be started with drills, routers, even contractor saws. Then fine tuned with gouges, chisels and scrapers. Very rarely will you need a gouge with a sweep grater than 7 or less than 4. 8mm-18mm widths on both chisels and gouges are used most anything over 25mm sits in my rack unused.
Fine tuning and final shaping would be rasps, files and scrapers again ... chisels and gouges being used to inlet a part or fine tune a line.
So what kinds? All kinds - My most used chisels are 8mm & 12mm straight fishtails, 18mm, 10mm & 5mm straight chisels #3, #4, #5, #6 & #7 straight palm gouges all 10mm - all Two Cherries or Pfiel - 2Cherries edge stays sharp longer, Pfiel has a better selection of geometry & palm handles are more comfy
But specifically for what you are trying to do... I probably would not use a chisel. ~ see my last answer.