Where do you want to be published? Big-5 and other major publishers/imprints require agents if you want to get in the door. If you want to be published there, practically speaking you need one. Even though a few might say you can send unagented mss., *realistically* you need an agent. The vast majority of material that they buy is agented, simple as that, and "open submission" periods that some offer seldom yield much of anything. Why make it any harder on yourself than it already is by not having an agent.
OTOH, of all the categories in kidlit, PBs have the best chance of selling without an agent, and if you meet an unagented author who's doing well getting traditionally published, chances are good that they write PBs. Do keep in mind, though, that some of them got started before agents were as necessary as they are now, and they continue to publish based on their track record and editor contacts. A new writer's situation doesn't compare.
Yes, query only one PB, but have at least 3 polished and ready. You won't talk about the others in any detail unless you have a phone call with an agent.
If you want an agent, pursue agents first. You can always look for a publisher if you don't get an agent, but you can't exhaust a bunch of publisher possibilities first and then query agents. They won't pick it up because it's shopped.
As for traditional vs. self-publishing, choose one career path and stick to it for now. If you want to be traditionally published, it's a long haul so don't give up "too quick." If you want to self-publish, that's an entirely different business decision and not something you want to jump to just because you got impatient with your first goal, or because you think it's a shortcut to getting a traditional deal (it will almost certainly prevent any traditional deal for that book). The self-publishing boards here are a wealth of information. Some authors become hybrids -- having both traditional and self-published works -- but almost never for the *same* material.
In the children's market, self-publishing is hard to succeed at for anything younger than YA.