Help for Clueless DM

The easiest way to introduce the players to the planes is to start with the planes themselves. Invent a small planar plot that would require the PCs to see a few VERY remote locations, then after that adventure, distract them with some stuff on the Prime. Then, lead them to a Planar Pathway for an adventure that takes place there, thereby opening up the planes to them. Introduce the factions by letting them meet faction members.

The worst way to start a planescape campaign where everyone is clueless is to throw them into Sigil straightaway. For one thing, you won't have enough experience with the PLANES to DM a good Sigil. For another, it will be TOO MUCH INFORMATION for the PCs to absorb meaningfully. Lastly, a few sessions of explaining sucks compared to starting on the planes having a bunch of good adventures where explanation takes a back seat.

I started one planar campaign in quicksand. That's right, quicksand.

The PCs took the bait on one of those 'take out the goblins who are stealing our hunters' 1st level plot hooks, but on the way to the Mayan-like Temple that they roosted in, the whole bunch gets caught in some quicksand. I have them make some desperate rolls, make them think I'm just as much a victim of the rules as they are, and they just keep sinking down and down. The goblins see them and try to fight them from a distance. One of the PCs manage to pull one in. One after another they sink below the surface. Everyone is sore, thinking that I just killed their characters in the first two hours of the game.

And then I said something to the effect of, "As you sink deeper and deeper, convinced that each lung-full of sand will be your last, you feel what you think to be the bottom... And then that gives way as you sink into even firmer and firmer dirt. Pushing beyond that, your eyelids filled with soil and grit, your body paralyzed by the ever thinkening earth, you begin to feel a warmth. Suspicious that you've died and gone to wherever, your quite shocked as you slide out what is now rock and onto the bare floor of some subterraining passage. There are little folk here, mole-like critters with big buck teeth... They're digging their picks into the tunnel walls... And if you can believe this: they're mining through solid gold to get to little chunks of rock. You clothes sparkle with gold-dust." And so began their first planar adventure on the Elemental Plane of Mineral.

They returned to the Prime thinking that the Outer planes are just tainted mixtures of different inner planes, that the factions are a bunch of barmies with no common sense, and the Lady of Pain is a transcendent Shad. Boy were they surprised on their next visit.

I'm on a similar course to you; I've got a party of 4 primes that I have intended to send to the planes since the campaign started. Originally I was going to get them there within a couple of gaming sessions but we've played about 8 9-10 hour sessions and they're still on Faerun(!) - the plotline has been keeping them there. This will be my and the players first planescape campaign as well, so my approach to it may be of some help to you.
What I've been doing is sewing a lot of planescape stuff into my campaign - I'm planning on a lower planes focused campaign, and at the very start I gave them a powerful staff that I've been using as a hook. As they uncover the staffs powers one by one I've been introducing the Yugoloths, the night hags and larvae, tanar'ri, and the inter-relations between these races. I've introduced some characters such as Shemeska, Estavan and Rule-of-Three (both directly and indirectly), organisations like the planar trade consortium, the black cult of ahm, and of course Sigil and a couple of factions (via NPCs that travel between Faerun and Sigil). So I've been trying to give them a very basic (and limited) view of the planes so they aren't completely lost, but are still clueless enough to be wowed by Sigil and the planes.
My plan is to have them encounter an enemy at a portal, as they flee through the portal the enemy will change their destination by using a different key, and they will find themselves at the fringe of a blood war battle. They flee (I assume they'll flee, not keen on a TPK straight up, they're only level 5!), and find a portal to Hopeless which gives them a nice starting base. Then a trip across the outlands before finally arriving in Sigil, and the real meat of the campaign.
Of course this relies on having a fair bit of resources handy so you mileage may vary.