I start my seeds in the containers you get rotisserie chickens in at the store. We buy a roasted chicken once in a while, they have a nice solid plastic pan for the base and a clear dome cover. Just about the right size to start two packages of tomato or pepper seed in each one.
When they set true leaves I move them up into plastic cell 4-pack trays. I set the trays in a standard 10x20 nursery flat, fill them with potting soil then flood the tray with water so the cells are literally filled with soggy mud. Then I flood the starter tray as well and let it set so it really loosens up.
Any sort of stick or plastic spoon to loosen the soil in the starter tray then gently tease the seedlings out of the resulting mud. I poke a hole in the mud in the cell with my finger then push the seedling, with any soil that clingss in the roots, into the finger depression. If the seedling comes out bare root I just drape the roots over the hole and press them down as far as I can. Pull my finger out and sort of puddle the mud back in around the stem and it's done. Maybe ten seconds a plant once the trays are prepped.
This year I put about 100 broccoli and cabbages in paper pots, 40 peppers, 54 tomatoes and 24 cauliflower in cell trays and so far 0 lost seedlings. There is one pepper that may not make it, it was barely sprouting, the seed case was still on it so it is chancey. Fourth year I have done it this way and never any appreciable plant loss.
They don't get potted up again, from here they go outdoors for a week or so here at home then they go to the garden and into the ground.