I've been using a utility knife (making sure to use a sharp blade), painter's 5 in 1 tool, and a couple of putty knives to cut/peel/scrape the old caulking out. Luckily this caulk isn't super old so it's not all hard and dried out. It's fairly easy to cut and pull the old caulk out. The hard part for me is trying to cut the caulking from the wooden trim side - trying to cut close, but not cut into the trim. I have managed to take a few shavings of wood trying to get a little too close.
A chisel - that's a good idea. I never considered trying to use a chisel for this.
I'd never have thought to use a multitool to remove old caulk. I mostly think of them being used to undercut a door casing/trim when installing new flooring - I guess they have a lot more uses, hence the name multitool.
I usually just use a razor knife and hold it so that i'm cutting a slight back bevel on top of the trim. That way if i dig in and take a little wood, the new caulk will be filling the gap anyway. Then I come back with a the razor and use a little more finesse to clean up any residual.
I've used a multitool in the past, and for me I usually end up doing more damage than good. I have a tendency to want to go too fast.
Oh maybe that's part of my problem. I suppose the same is also true for a top coat of paint? Last time I didn't bother to paint over the caulk since it blended in fairly well to the white trim.
Without primer the caulk does not bond well. I believe it is because of the rate it loses the moisture at. The tube states to apply over primed surfaces.
Top coating is up to you. I prefer top coating because it is easier to wipe dust off from the painted surface.
Reading instructions is over rated [wink]. It never ceases to amaze me what I can learn when I go back and read the instructions later - "oh, so that's what that's for."
Now you that you mention it, the dust does seem to stick and kind of embed into the caulk.
I usually strip, sand, prime, caulk, prime again, then topcoat. If the trim is in good shape I may not go with the second primer coat. I don't mind the extra time since I only do this for my own home and not for a paycheck.
I finally got around to applying the first round of Big Stretch Caulking. The application and cleanup was a piece of cake. The first flight of steps that I re-caulked look great to me. I'm actually really happy that I don't have to see that nasty old cracked caulk line every time I walk up the steps. Now I just need to re-do the caulking on the next 2 sets of steps/floors.