Anyone who has ever planted a pawpaw knows about the taproot, but I still get asked quite often about how to plant small pawpaws, and I still hear from people who loose pawpaws when they try to transplant them. Someone recently brought it up, so I figured I'd provide a few links that might answer questions for any potential Tennessee Pawpaw farmers.
I always recommend planting pawpaw seeds at the location where the tree is desired, rather than trying to grow the seed in a pot and then take the chance of loosing it after transplant. One instance where that is not feasible is when a pawpaw cultivar is purchased already grafted to its rootstock. In that case, transplanting is necessary.
When I collect pawpaw seeds from my pawpaw orchard, I always make sure they are fresh seeds, thoroughly clean them, float test them for viability, and place them into cold/moist stratification for storage. Sometimes I use paper towels sprinkled lightly with cinnamon (for an attempt to eliminate/reduce any fungal/mold growth) as a stratification medium. I have had excellent results so far from doing it that way. I also keep a close check (every few weeks) on moisture levels in the plastic bags, because the refrigerator can suck all the moisture out of the bags pretty quickly if there are any holes in the plastic. Pawpaw seeds are recalcitrant, BTW, so they go bad relatively quickly if allowed to dry out for more than a day or so. I keep the seeds in stratification until early spring.
When planting the seeds, I plant them about an inch or less below the surface and cover with a light layer of mulch. No amendments are necessary, and I wouldn't suggest using any. It is important, in cases where pets or wildlife may be present, to protect the seeds from predation.
Pawpaw seeds typically take a really long time to germinate. Germination around the end of summer or even later is common. The seeds will develop a decent sized taproot before sending up any foliage, so don't give up on your seeds too early. Even if you don't see any signs of life, you still need to ensure that the soil is moist even down to, at very least, a foot deep. Don't make a swamp, but do water kind of like you would with a newly planted sapling tree.
Pawpaws are more productive in full sun, so I would recommend a permanent location with at least mostly full sun. The trees are found in the wild as understory trees, so they will do just fine in a shady area but won't produce much fruit. One important consideration is that the trees really need partial shade for the first couple of years of their life. Small seedlings can easily cook in full sun. I recommend providing them with a 50% shadecloth covering for their first two years. Remove the covering in very early spring (before they leaf out) of their third year. A single layer of loosely-woven burlap or some other material can work if shadecloth is not readily available.
Probably the most widely known and respected center of pawpaw research is the Kentucky State University Cooperative Extension Program. Here's a link for part of their site that discusses many aspects, including the extra care needed to deal with the taproot when transplanting a small pawpaw:
Here's a more technical paper that discusses various aspects of container production of pawpaws. It has a nice picture of the huge taproots present on even very small pawpaw seedlings: