If it makes a difference, I live in central Arizona. So, it's already quite warm.

I have some seeds growing as well. Plant them in rich potting soil. Lychee seeds should be laid on their side, and planted about 7/8ths of an inch deep, the soil covering it should be loose and kept slightly moist. It should be put somewhere with sun, but not full sun. You do not want your seedlings drying out or getting scorched.

The seeds should be laid on their side because the sprout and root come out of the same end.

While lychees are easy to grow from seed, they can take 6-12 years before they begin producing fruit. There is also the issue of fruit quality because, like many other fruits, when you grow lychee from seed you can not be entirely sure what traits you are going to get from that plant, it may produce less desirable fruit, although lychees are not as bad in this respect as many other fruits. Lychees are usually propagated from cuttings, both to ensure reliable fruit quality, and because it cuts down the time it takes to produce fruit to 5-6 years.

I am really not sure why lychees are not more popular. Their taste and aroma is almost like roses, and there is only one other fruit I am aware of that is like that—raspberries—and they are one of my favorite fruits.

If your seedling looks like it is wilting and you are in a hot climate, it probably needs less sun, and possibly more consistent watering (every 2 days). Giving too much water is not going to help, the important thing is frequency of watering. The main thing is the soil should always be kept slightly moist. I emphasize this because some people reading this live in dry climates, and that can be an important issue with tropical plants. But do not put the seedling in too much shade either, it does need some sun.