Any decent acrylic general purpose primer will work for this.

So I don't need something like Gardz?

Not after it has already been painted. Myself, I would remove the whole mess, but thats just me,I hate painted wallpaper for just the reason you mentioned and it is going to get nothing but worse in the future

Well, it's not an option at this point to remove it. We have wallpaper in all the rooms in our second floor, and the walls would need to be rebuilt behind the wallpaper. We can't rebuild all our walls on the second floor right now, for financial and other reasons. We plan to do one room at a time eventually. For now, unfortunately, we have to make the best of it.

Can you recommend a good acrylic general purpose primer? (For what it's worth, the person at my local paint store said I should use Gardz, which sounded a bit like overkill to me, especially, as christophern noted, the wallpaper is already painted. But I haven't been able to find any good information about whether Gardz is appropriate for surfaces that are already painted).

Whenever I go to prime, do I spot prime first over the joint compounded areas, and then go back over the wall? Or should I just do the whole wall in one go?

Thanks for the advice!

Gardz is not made for going over paint, as paintguy alredy said, any quality primer will work and do the whole wall after the patching process. Also, this will involve a trip to a REAL paint store not blowes or homely depot.

Thanks very much for the advice. Interestingly, it was someone at my local Benjamin Moore who told me to use Gardz; person at Home Depot told me to use a latex primer (Kilz 2). Should I use something from BM, like Fresh Start, or is Kilz 2 okay? I'm also thinking about 1-2-3.

Just so you know this is a result of not SIZING the wall before the wallpaper was put up. I realize you are probably not going to re-paper, though.

I recommend BIN 1-2-3 [Znisser] as a primer for your situation.

Not to pick a fight with Jump, but 'sizing is the WRONG term. Here is an article that explains it all

Here is a link that might be useful: primer vs sizing

. Should I use something from BM, like Fresh Start, or is Kilz 2 okay? I'm also thinking about 1-2-3.

Fresh Start, good, 123 good, Kilz2 not good