What does a meditative state feel like?

I’d like to learn how to meditate, but I’m not sure what to look for. So far, I’ve been trying to get lost in images. I picture the night sky, or a meadow full of rich, earthy scents and sunshine – you get the idea.

This does relax me, but I can only concentrate on any of my images for seconds at a time. My mind goes to tasks, plans, or a meta level, where it applauds my meditative state, while, obviously, not acting very relaxed at all.

Once or twice, I felt very close to falling asleep, in the sense that my mind went blank, and my body felt heavier. Is this what I should be looking for? If so, does that mean I should be doing my meditating sitting up, to keep from falling asleep?

Bonus question: Does it make sense to set an alarm to a specific time, and fight against my desire to jump up after only five minutes? If so, how long should I be aiming for?