Why do I start dreaming so quickly?

Example: I feel very tired, fall asleep for a nap around 3:30, wake up for some reason around 4, and by that time I have already had a dream. Or, more often: I feel very tired, fall asleep, dream or don't dream, wake up and do something that requires full consciousness (like answer a text coherently or walk across the room), fall back asleep, and start dreaming very quickly. I'm not sure if I'm like this at night; it's harder to tell, since I don't usually wake up so soon after falling asleep. I always thought that dreams happened in the REM stage, which is usually about 1 to 1 1/2 hours into sleep. My dreams are "normal," though I might remember them more often and in more detail than most people I've talked to. Why am I like this? Does it have to do with being tired, or taking a nap? Are my cycles so different? Do dreams happen outside of REM?