Do you really want to paint this piece, if it is hardwood??
This is just my opinion, I hate to see nice pieces painted. I would restore it instead.
However, if you are painting it.
1. Remove all old paint , either with fine sandpaper (going with the grain of the wood) or paint stripper
2. Rub the complete item down with steel-wool
3. Rub the complete item with turps. to remove all dust etc.
4. Apply a good quality undercoat paint with a soft brush (not a nylon one- as it will leave streaks)
5. Allow to dry well
6. Apply finishing coat of paint with soft brush
7. Allow to dry completely, replace knobs/handles
And 'Hey! Presto!!' A completely new chest.
I believe that if paint is diluted with white spirits or turps. 2 light coats are better than 1 heavy coat & always paint in the same direction as the grain of the wood.
I'm sure that you will have others with more suggestions.
And MOST important, WASH & DRY you brush, before re-using.
Bunty!!!!! Good-Luck with project.