I have strange dreams all the time, but generally only remember them if I think about them immediately after waking up. If I sit there and think "gee that was weird" and think about what happened in the dream, I'll remember much of it; if I go straight to the shower or don't try to remember what happen, then later if something comes back to me, I'll only remember a small bit of the dream.
Most of my dreams seem to be of people from my past juxtaposed with people I know now. It usually involves being in marching band, either in high school or college (real experiences.) If I'm dreaming about high school, there are people from college in it, and if I'm dreaming about being at UK, it usually involves people from high school. Add in a few people I work with now or in the last few years, and it's often just a weird combination of people, many of whom I haven't seen or thought about for more than a decade, especially the one I had a couple nights ago.
It seems like whenever I awake from a dream, I always think "that was weird" and never seem to think any of them are "normal."
I do sometimes realize I'm dreaming, and that's kind of fun, though I usually wake up shortly after that happens.
And sometimes, I wake up after one dream, go to the bathroom or get a drink of water, go back to sleep and have another dream.
(And btw, didn't Louis Armstrong sing "dream a little dream of me"? At least that's the version I know.)