The absolute minimum you need to do when planting using biodegradable cups is to remove the bottom. Even though these cups are designed to biodegrade quickly, it still takes months, and will prevent the roots of your plants from expanding into the surrounding soil. Just gently tear off the bottom, and loosen the roots.
Though many say it is not necessary to remove the remainder of the biodegradable material, I always peel as much off as comes off easily. Sometimes the roots have begun to penetrate this layer and only small portions come off, but sometimes (as shown above) the entire casing will fall away when gently pulled on. Again, just lightly loosen the outside roots. Then, simply plant in hole even with surrounding soil.
A few weeks after transplanting you'll need to start to sucker (or prune) your tomatoes. You may think that the more vines your plants grow the better, but this isn't the case. Most tomato plants will continue to grow their vines through the entire summer. All those vines take a lot of water and energy to maintain, and that's water and energy that won't be put into producing fruit!!
When 'suckers' form, so named because they suck production away from the fruit, begin plucking them off. You can use small scissors, or just pinch them with your fingernails. You'll know the suckers because they'll be small vine sprouts that form in the 'armpit' of the main stalk and auxiliary stems.
Be very careful though, because early blossoms and often look like these suckers and if you remove those as well you'll end up with a bare tomato plant!
As you can see highlighted in purple above, the suckers come directly out of the 'armpit' and have leaves exactly like the rest of the plant. The tomato buds highlighted in orange, while close, do not come out of the armpit, and have small teardrop shaped buds.
While it's best to remove the suckers early, do so ONLY if you're confident you can discern the difference between the suckers and the buds. Otherwise, play it safe, and wait a week or two and the difference will be more noticeable.