I was curious how people speed things up for both types of cubes. After browsing for a bit, I've decided the approach best for me is:
- Reduce to 3x3x3 case as described by Chris Hardwick. Now standard Rubik's Cube algorithms can be used.
- Solve the cross.
- Solve the first two layers except for one corner piece. Originally I had planned to follow the "ZB" system, which has you solve the first two layers except for one corner piece and one adjacent edge piece, but this leads to numerous cases for step 2 of the system. For now, I'd rather sacrifice a few moves to get one more edge piece in place to cut down the number of cases for the next step.
- Perform step 2 of the ZB system to put the last first-layer corner piece in place while at the same time orienting the last-layer edges. They are the last few cases on this page. (I don't think I could ever memorize the entire ZB system. I lose a few turns putting the last middle-layer edge in place, but I only have to learn a small fraction of the cases.)
- Either orient the last layer ("OLL"), and then permute the last layer ("PLL"), or for more efficiency, achieve both goals simultaneously.