2. Hang the thing. Yes, there is a semi-science to the art of getting the height of a piece just right—it's called measuring. To be exact, the center of a framed piece of artwork should be 57 inches above the ground (that being the average human eye level, and the height galleries and museums use to decide where to hang pieces). Mark that height using a pencil, then measure to find the middle of the wall (from side to side), and mark where the two points meet. That's where the middle of your artwork should go!
If you're not up for hammers and nails, just lean it. The easiest way to display art is also best for anyone who is afraid of putting nail holes in the wall: lean the frame against the wall, or on top of a piece of furniture, or on a shelf somewhere.
If you're always re-arranging, consider a picture shelf. If you're into the whole leaning thing and want to formalize a place for such activity, consider adding a shallow picture shelf in one of your rooms. It's a perfect solution for those with constantly changing styles (or the rearrangement bug).
...Or a picture rail. If you're into the idea of sparing your precious walls from holes but want a more formal look than leaning, consider a picture rail: a sliver of molding that goes up near the ceiling, from which you can hang your art on hooks and strings—and then change it out whenever you feel like it.
Leave some pieces unframed. Maybe you've collected some of those paintings on boards from the flea market—lovely peeling edges and all—and want to preserve some of that charm without paying for a fancy floating frame. Or maybe you just want to hang up wispy paper drawings and call it a day? Leaving certain artworks unframed is completely fine, even encouraged. Just tack them up without fanfare.