One of the best things a DSLR can do that compacts can't is capture moving action, this is a photo of my daughter running at me flat out, generally even a good compact will struggle with this
That was taken with a 17-55 f2.8 IS lense and 40D in motor drive mode, I had 15 pics to choose the best from as she ran towards me.
This is taken with a prime, 85mm at f1.8, the background is nicely blurred, again compacts can't do anything like it
and indoors an external bounce flash can blow away anything you can do with a compact (quite a lot of PP on this too)
But for landscapes compacts can be quite good, this was taken with a canon G9 from upstairs first thing one morning with the DSLR in the car boot, probably would have missed it by the time I'd gone to get it
You just have to go and take lots and lots of photos and figure out what works and what doesn't, I've found the people here
very helpful for feedback