How can I get rid of large amounts of crabgrass without killing the good grass?

For a chemical-free method, use a pair of long needle nose pliers. Get under the central whorl of the crabgrass and yank out most of the roots. Start as early in the season as you can, and be sure to get the root and the central whorl or it will just grow back. A garden kneeler will help. Pick a high priority area and establish a zero tolerance zone there, removing every crabgrass plant in the zone. Be vigilant; don't allow even one crabgrass plant to thrive and throw seeds and tillers in that area. Expand the area year to year and you'll be surprised how effective this is. Remove as many plants with a central whorl from other areas as you have time for.