If skunks are digging holes in your lawn, you may have a grub problem. Skunks are a natural enemy of white grubs and won't hesitate to tear up some grass to get to them. If this is so, the best way to combat the skunks is to control the grubs.
The best way to get rid of the skunks is to deter them from entering your yard. This amounts to removing their food supply (your pet's food and water supply, the garbage, or invertebrates in the lawn) and/or fencing. Trapping is an option, but there is the odor. It is not recommended that untrained homeowners try trapping and moving a skunk.
Remember that if the habitat is attractive to skunks, it will be attractive to future skunks as well. Make your lawn and yard as unfriendly to skunks as possible, and see what happens.