One way to help keep all the parts from sounding like the same guitar is to change up the way you record them. For example, record the clean parts direct, and maybe apply the needed effects (if any) after the fact. Maybe record the "chunky" backing chord parts that way using amp simulations. Then you can set up and mic your real amp with stompboxes to record the lead part. Or switch everything around if it sounds better another way. The main thing is to keep it from sounding like one guy playing three parts on the same guitar.
I like to record lead parts with all the effects done live, but not everybody does it that way. I'm just more comfortable playing that way since I can hear how the instrument is interacting with the electronics in real time. Trying to apply a lot of creative effects to the lead part after the fact is usually a train wreck for me.
What do you mean "make it cut through the mix"? I've been trying all afternoon to bury it with a stake through it's heart!