Is there a program for Mac that I can use to copy DVDs to iTunes? I've tried Handbrake but the quality is poor. Maybe there's just a setting I need to change?
Given that you want playback on your Mac and iOS devices, Handbrake is really the way to go. As you note in the comments, it's not lossless, but you should be able to get pretty good quality, and with a good reduction in file size.
If you find the quality of your first attempt unacceptable, you can adjust the settings to improve it (typically at the cost of file size).
A good starting point is the Universal or iPad preset in Handbrake. The RF slider adjusts how much compression is used (higher RF = more compression = less quality).
Try setting it around 19, maybe 18 (it's logarithmic, so small amounts make big differences) and do an encode, see how you feel about the quality. If it's still lacking, reduce the RF a bit more and try again. Perhaps just pick a chapter or two as a test encode until you figure out what settings you want.