I always use oil paint. I've tried several latex kinds and never found one that would wear as well in the long run.

For primer, I use a spray. It's fast, no brush marks and dries super quick. For small jobs such as one or two pieces of furniture, one I particularly like is Rustoleum auto primer (dark charcoal gray if I'm painting dark and the light color if I'm painting light).

I should add, there is a spray primer that is absolutely awesome, hands down the best, but it's expensive and I usually don't have the budget for it. It's XIM brand, and it's a UMA primer. The rustoleum has worked fine for all the furniture projects I've done (and that's been a lot) but if this were a heirloom piece or something you didn't mind spending a couple extra bucks on, you might try it. Some Sherwin Williams stores carry it, but you may have to order it. The paint on kind is great too but you have to paint it on - the spray is magically easy.

Yes, an oil based or shellac based primer would have been best. You can do a light sanding and then prime again. If you do another coat of paint, most likely you'll have the same issue.
You can paint oil over latex, not latex over oil. But those rules don't apply to primer.

Sand and repaint with the oil based primer. Do anything else and you will regret it.
If the wood shows thru the primer NOW it will show thru in short order thru another coat of primer and the paint.
It's extra work yes BUT in the long run if the prep work isn't right the piece will never look right.

Thanks everyone. I went and got the spray primer. DH worked on it a little and the footboard blew over and landed on the grass. Now it looks worse. :-( Plus, some spots are thicker than others.

I think I'll wait until spring time and use the liquid sander to strip all the paint and primer off and start from scratch. There's a lot of intricate moldings on the footboard and headboard and I don't want to lose any of the details by having too many coats of paint and primer.

paintergirl94 is right, use a shellac based primer. I found out about that stuff the hard way. I used white latex primer and off white top coat on all my crown molding, base boards, window and door casings in the bedroom. Yep, couple weeks later I had all these orange spots coming through (except on the crown molding thank goodness)! I was frantic.

I had to go over every orange spot with the shellac based primer, mostly I ended up painting the entire affected pieces of woodwork with another layer of off white. So far so good.

Forgot to mention, it was the knot holes in the pine wood coming through causing the orange spots.

I painted my pine stair treads and used Zinsser's BIN shellac based primer. It's for sealing knots. Works very well!

Well, the original wood underneath is a burled walnut (I know, I was very hesitant about painting over such a beautiful wood, but parts are it are fairly damaged and I didn't want to restore the finish), so it's not the knots that are causing the problem.

Any advice or comments about how to strip down to the wood? Is liquid sandpaper any good? Or should I just go with a paint stripper? Or is it the same thing?

You're probably seeing the tannin of the wood seeping through. Or, not all of the stain was removed. Either way, it's because water-based primer was used.
I would suggest going to your local paint store and taking a look at the strippers available in your area. I have found gel strippers work best. You will also need to use sandpaper and steel wool.
Liquid sandpaper is to use for *etching* a surface. Usually when you want to paint a glossy surface.
Zinsser makes excellent paint strippers.

Thanks paintergirl!