I noticed how a coffee sleeve resembled a small flower pot. Since it was tapered I thought It might hold soil, especially after the roots were established. I was surprised how well it worked so I'm sharing.
Step 1: Get the Sleeves
You might save your own sleeves or collect them at work etc. The point being this part should be free and reduces/reuse waste. A lot of "raw" paper products decompose nicely into rich soil. Don't use color printed or glossy stuff, consult the interwebs for more detailed advice, but suffice to say it's safe and beneficial to use paper in the garden.
Step 2: Fold Them Into Squares
I folded the sleeves and made them square. Square pots fit together making it easier to pour the potting soil, plus they save some space. Now place them into a tray that will hold water, I used the top of an old seedling starter tray.
Step 3: Fill the Pots
Fill the pots half way and lightly compress the soil to lock it in place. Don't press too hard or your plants might not grow well. You won't be lifting them for a while to let the roots grow through so it's ok if it's a little loose.
Step 4: Transplant Your Seedlings
Gently place your seedlings in the half filled pots, sprinkle the loose soil in around the plant. Don't press the soil around the plants, instead water it and sprinkle more soil if needed.
Step 5: Wait for It....
Now all you have to do is let the plants grow. Mine reached planting size in a couple of weeks. When I watered them I left some water in the tray to ensure the soil was completely wet. Some soil will leak out, leave it in the tray and it'll help hold moisture around the base of the pots.
Step 6: Transplant Your Veggies
Once your plants are big enough it's time to put them in the ground. For my tomatoes I bury them up to the first set of true leaves. Some plants don't like this so consult the interwebs for proper planting guidlines. It's important that you bury the entire pot, exposed edges might wick the water away from your seedlings. If needed, trim the top edge off or cover with mulch. Don't try to remove the pots, you might have roots growing into them. The pots will decompose and help feed the plant. Happy gardening!