We've recently replaced our kitchen tiles. I'm happy with the results but have no idea what to do with the edge. How do I clean this edge up so that it looks more appealing?
If you don't want a visible transition strip, you could also put down a tack strip right at the transition point, attach the carpet to the strip and then trim/tuck the excess into the crevice between the tack strip and the tile edge.
Only real "difficult" bit is to ensure the carpet and tile height are even so the carpet fibers will effectively conceal the cut edge of the tile.
You'll want to install carpet to tile transition strips (sometimes known as edge trim). Here is a listing from home depot. Basically it's a decorative metal strip that goes over the carpet/tile gap and provides a nice, attractive transition between the two.
Here are some instructions from eHow on how to install the transition.