For the very early seedling sets that I can put in the garden early, I like to use newspaper, coir, or some other natural pot that I don't have to remove the seedlings from. That way I can just plant the entire pot and not have to worry about what to do with the remains. The material will go back into my garden for enrichment and I don't have to throw anything away or store anything for the next planting!
However, for the plants that I will keep indoors a little longer, like my okra or winter squash, I use plastic cups. The reason is that I will have the plants in the starter pots longer, and I was having trouble with the pots breaking down too rapidly to use for long term seedlings. Those little cups that fit in the single use bathroom dispensers work pretty well. The 3 oz. ones work okay, but I generally use the 5 oz. ones because it gives my little roots more room to grow and be a better base for the plants I put in the garden. I never use the "Dixie" style cups, the cardboard with wax lining, because they don't break down very well in the garden after I've planted, and they do not last for more than one use before the moisture ruins them.
I just use an awl to punch a hole somewhere in the bottom of the cups. That way I can water and not worry about oversoaking the seedlings. After I have emptied out the cups into the garden I just put the cups away for next time. Some people would say sterilize to prevent contamination, but I have never done this and have never had a problem with that from year to year. They stack together very nicely and don't take up much room in my work space.
One other thing about the cups I use - if you leave them in direct sunlight for too long they can become brittle. But these are cheap and easy to come by, and I can usually use the same cups for two or three plantings without a problem.
Sorry - have to get back to planting some more seeds. Happy Gardening!