Welcome
to ATP and my sympathies on your slum lord status.
We've all been where you're at.
Thinning is the hardest thing to do when you first start out with seedlings. Here's what I'd do:
First get your upscale accommodations all ready to receive the baby plants. Fill it up and get the medium moistened thoroughly.
Sit yourself down at a table with good light, a small jug of water, and some paper towels. I use my kitchen counter for this job because you're going to get it a little messy, probably. You can put down newspaper to work on, but I always forget to do that. Let's say you're going to try to transplant 3 cells of each kind of veggie - you have 6 kinds, right? (you didn't tell us what else is in the 144 pack that hasn't come up yet.) At some point you need to figure out how many plants of each you really want, and thin down to that many.
Find a small, thin tool something like a bamboo skewer you'd use for shish kebabs, a metal nail file or a plastic disposable knife with a very thin blade (there's an article about using a palette knife for this job, but I'd be afraid of cutting the tiny stems of the plants.) Once the medium is nice and moist, it will 'hold' if you make a hole in it ready to receive a baby transplant. Make two narrow, deep holes at least an inch apart, more like an open slit in the medium of each cell.
A dinner fork will work well for extracting one cell worth of seedlings at a time. Slip it down the side of the cell, and gently lift out the whole cell, seedlings, medium and all, and lay this on a damp paper towel or newspaper. Work quickly to choose the six strongest seedlings - you don't want them to dry out. Tease them away from their buddies, and delicately thread the roots down into the holes in the new cells so you have two plants to a cell. Place them so the soil is at the same level as they were before. Yes, later you're going to have to choose to take out the weaker one, but this is 'insurance' in case only one survives the transplanting. Push the holes in the medium closed around the roots with your finger or the thin tool.
Once you've done this with all six types of plant, the remainder in the crowded slum you should probably just thin to one or two plants per cell, and see how they fare. You can cut off the crowd with a tiny scissors, or pull them out, holding your finger against the ones you're going to keep. Tease them out carefully and slowly to not tear any roots, this surely takes patience and dexterity. Maybe a small tweezers?
Water everyone with a very gentle flow from a small container onto the medium. OR a spray from a spray bottle can work, but might make the separated seedlings too heavy, you don't want them to 'flop' or the tiny stems might break. Direct the spray below the leaves to just wet the medium. Keep them in the shade until they look like they're recovering then gradually ease them back into good light.
I've done this a LOT lately at our school garden - try telling kindergartners to spread out their seeds! So we dig up the clumps, get the kids to make holes in the soil with their fingers and hand them a tiny seedling to plant. (we do the separating of the crowded clumps) You'll be surprised how many seedlings make it through all this rough treatment! Yours will do even better.
It will help them a lot if you can keep the humidity fairly high around the new transplants for a day or two. So keep them on the cool side and away from heat vents and drafts. Maybe spread damp towels or placemats around them on a counter at night?
with everything that was said. 
I hate to say it but I kept most of them and now I need to buy a few (lot) more trays of the larger pots. 
Looks like I'll be feeding the entire neighborhood this year. 
