Seedlings in plastic cups should be transplanted into larger containers of nutritious soil after the single-bladed leaves reach their full size and the 3-bladed leaves start growing in.
There are many different opinions for which potting soil to base the mix on. We use Edna's Best potting soil and we've had fine results with Black Gold and a few others. Some garden centers have a display with little trays of each soil the carry, where you can see and feel the texture of each. No soil will provide everything the plants can use all the way through the end of flowering. As long as sufficient ferts are applied manually, all the soil needs to do is drain properly, hold enough but not too much water, and support the plant. Amendments like Perlite, Pumice or sands are added to (amung other reasons) increase drainage & allow water to spread more easily throughout the soil. They are used when the soil mix is too spongy and they break up the tangled mess of fibers. We add about 15% extra perlite to the potting soil.
Bone Meal and Worm Castings are also added to all of our mixes. 20-25% extra Worm Castings should go into most soils. There are other additives we use but those are the most important.
Hope this helps.