I have two batches of Amaryllis seed sprouting, one on water and one on potting soil. I can't say for sure yet, but I hope the ones on potting soil are sprouting like the ones on water. How many years should I expect it to take for them to make first blooms?

What soil/nutrition is the best possible for their health and growth?

It is depending on your weather, if it is hot enough in winter, so plants will not be dormant in winter. They need a lot of heat, not not direct sun (if temperature is over 90 F). In ideal conditions you need about 4 years.
It is depending on the Specie of amaryllis as well.

Amaryllis love soil very each in nutrients , I usually use regular soil from nursery that based on coconuts . I feed it ones in 1.5-2 month with regular granular fertilizer. I found that it can grow even on 100% horse or cow manure (mature plant only, not spouting seeds) .

Thank you Ron. It is most definitely hot here in the summer, and in the winter the temperature rarely goes below 22*F. Maybe I can select places that will get some shade during July-September.

22F is too low. From my experience, if temperature goes below 45F for several days, bulbs will loose all leaves (specially the blooming bulbs)

Usually 3-4 years to bloom. I have found mine grow better if I add some weak liquid fertilizer about once a month.