That's such a tightly framed picture that it's hard to tell for sure, but since you said you removed the trim, I'll assume that the window is on the left and the wall on the right.
The normal method for filling the gap between the window and the framing is to use minimally expanding spray foam. This will expand to fill all the gaps, but won't expand too much and risk warping the side of the widow casing causing it to bind.
Once you've got the foam sprayed in to fill about 1/2 the space, let it expand and cure. Add more to fill in any gaps as necessary and cut off any that sticks out beyond the frame/wall, then reinstall your trim.
If this is the exterior side, then yes, you'll want to apply a good quality exterior caulk around the trim to keep water from getting behind the trim. You'll want to do the top & both sides, but leave the bottom open just in case any water does get back there - it'll give it an easy path to exit instead of sealing it into the wall. You say you're in Colorado where there's a tendency to get a fair bit of snow. If the window happens to be low and on the leeward side of the house, you might get drifting there, but it's not too likely that the snow will push up against the house and actually push into the gap at the bottom of the window.
If this is interior trim, then you only need to caulk it against the wall if there are unsightly gaps that need to be filled with wall/trim colored caulk just to make them less visible. The foam should be more than sufficient to stop any drafts getting through and will thoroughly insulate the opening.
Based on the added picture, I think what we were looking at in the closeup is this circled gap:
If that's the case, then yes, I'd simply caulk between the edge of the sill and the 2x4 framing in the window.
You indicate that this is a very small gap, roughly 1/4". I agree, it's difficult to get caulk into a gap that small, but it can be done. However, I'd cut the tip of the tube at roughly a 3/8" bead size, then put in a thicker bead, working it with the nozzle to try to get some pushed down into the gap and leave a reasonable amount above. Then, while the caulk is still soft & fresh I'd install the side trim, pushing it into the bit of a blob of caulk coming out of the gap.
By bedding the trim into the caulk, it'll sit exactly where you need it to go and the caulk should create a pretty good seal in the gap and at the end of the board. If you wait until the caulk has set up, you'll have some amount of blobbing there that will either need to be cut away, possibly opening up gaps where you didn't get it pushed into the opening very well, or simply leaving a very thin layer of caulk that may not last long, or it'll push the trim away from vertical and not look very nice for your finish carpentry.
Based on finally getting enough info to answer the question (that'll teach me to answer incomplete questions)...
In this case, I would pull/cut off any caulk that's in the least bit loose and replace it with new caulk. Make sure you get a good seal between the new and old caulk.
Actually, since part of that old caulk is against the smooth vinyl window and the rest is up against a piece of 2x4 that's going to be hidden, I'd scrape all of it off and reapply a single bead of brand new caulk. Failure of the old bead in one spot indicates that it may well be ready to fail in other areas and there's no sense in tearing the whole thing apart in another year to apply another patch.
