I sow my tomato seeds onto damp kitchen towel inside a 'margarine' tub with a clear lid. This makes a mini propagator. The brand of spread that I use now does not have a clear lid so I intend to cover the tub with clingfilm. Then replace the lid, having first cut out the largest rectangle I can manage.
When I can see the root appear I carefully plant the germinated seeds in growing cells. I find that a damp cotton bud is best as it lessesns the chance of damage. The cell pack gets placed inside a unheated propagator and given a little water and lots of TLC until they are big enough to put into larger pots.
These pots are put outside if it is warm and brought in at night until they are sturdy enough to go into the cold frame.
If this seems like a lot of fuss I do admit that I have time on my hands and very large crops of tomatos.