Why are the simplest words the trickiest to use correctly? What ancient grammar god has punished us so? Do not feel bad if you get this one wrong! Just study up so you can smugly correct others in the future. Or so you can use it correctly in case anyone around you actually knows and can be impressed.
And the answer is... [drumroll, please]... a) I'm going to lie down for a nap.
There are so many examples of these two words being used incorrectly in our society that you honestly are being led astray by the world on this word choice. It's not your fault. You're just going to have to memorize these and ignore the songs, titles, and the rest of the world telling you the wrong word.
The technical reason for this answer is that "lay" requires an object. It can't do the reclining by itself, it needs someone to do the reclining for it. So "I go lie down" but "I lay down the baby": the baby is the object that can't propel itself—safely—from my arms to the crib. Tell your dog to "Lie down!" and tell your child to "Lay your books on the table." The dog can recline itself; the books cannot.
Maybe this will help:
There's an "i" in "lie"... I am able to recline myself, so therefore, I use the word "lie" for people (and animals) who are likewise talented.
"Lay" is similar to "put": "Put your books on the table." So if you can use the word "put" then you can use the word "lay."
I don't go put down for a nap; I go lie down for a nap.
Note that this is only for the present tense: what you are doing in the moment. Past tense gets waaaaay more confusing, and I'm still blaming some angry ancient grammar god. Just to make your head hurt, the past tense of "lie" is "lay." I know; it's sooo wrong to do that to us. But just trust me; that's how it is. Past tense of "lay" is "laid" which makes much more sense. And then we get to the past participle (the one that needs a "helper verb" as it was called in my elementary school).
I am going to lie down for a nap. I lay down for one yesterday as well. Actually, I have lain down for a nap every day this week.
Lay down your books. You laid them on the table yesterday. You have laid your books in that same spot all month.
If you'd like to know more, I recommend Grammar Girl's post on this topic; it goes more in-depth and includes a small cheat sheet. Who doesn't like a good cheat sheet?