I did newspaper ones several years ago. They worked well. Don't make too many layers...I had a few that I cut at planting. Kinda depends on the plant & how long you need to have them in their pots. I've used cups as Bay says & generally cut them off before planting.
When I used a cup, I would be sure the plant was moist but not too wet when I cut the cup off, to have a moist but solid glob of root. Then slipped it into a prepared hole. I have some containers to use for mini greenhouses -- Like the loaf cakes, or roaster chicken come in. I have also put seeds into dirt in an oblong plastic pan, outside, transplanted when ready.
We have so many different weather situations that a lot depends on that, alone. Here my season is fairly long but, for an early garden, you need started plants....if they are heat loving types. Cooler ones, peas, brassicas, etc. can go in ground & do fine for the most part. I suspect you TX people have same thing.
NH -- holy cow, snow and cold creates a real short season. Can't believe you'd ever get a tomato without a started plant.