Firstly, daylilies love full sunlight which is where they will thrive. They will grow well in partial shade as well, but it would be best if you choose one of these two options and not plant it in a shady area.
Your daylily will be sold to you as a bare root daylily. This is to prevent those bad bugs from moving garden to garden! If you get daylilies from another source, I strongly suggest that you take it, and wash it out prior to taking it home. The plants don’t mind this at all and you will get a plant free of bugs. You might want to think about doing this with other plants that you bring home into your garden, hosta's for example can also tolerate being washed out and planted as bare root!
When planting your daylily, you will first need to pick the location where you are going to plant your new daylily. Dig a hole that is big enough to spread out the roots, to ensure that you give it the best chance of establishing itself. You can then just put your daylily in the hole and cover it up with soil and you will be fine. Although I recommend adding Alfalfa pellets and a sprinkle of general fertilizer such as a 15-15-15 mix. Then, I give it a nice drink of water before putting the top layer of soil on the daylily roots up until you reach the crown (just above where the root changes into a fan). Daylilies love compost as well if you have some great compost to put in with the new daylily.
It really is as simple as that! The best thing you can do for your daylily is to make sure it gets water and sun. They are not heavy 'feeders' but giving them a light sprinkle of the 15-15-15 mix every year will keep them happy as well.