Skunks and raccoons often damage turf as they forage for grubs and other soil insects. Generally speaking, skunks dig precise holes. These holes are shallow and may be very numerous, they usually are not "large" (1.5 to 2 inches in diameter). Raccoons often shred the grass and will even peel back the sod, exposing large areas of dirt.
The best way to get rid of the skunks or raccoons is to deter them from entering your yard. This amounts to removing the food supply (your pet's food and water supply, the garbage, and sometimes the grubs in the lawn) and/or fencing. Applying grub control after the damage begins will not help you during that growing season. It is not uncommon for grubbing to continue for 2 weeks or more after the initial application. Grub control works best when applied BEFORE damage begins.
Cage trapping is an option, but you will need to learn how. Once the animal is captured, relocating it may not be legal nor humane. Trapping requires a great deal of responsibility on the part of the trapper. Traps must be checked every day, including weekends, holidays, and in every kind of weather.
Remember that if the habitat is attractive to skunks and raccoons, it will be attractive to future animals as well. Make your lawn, and yard, as skunk and raccoon-unfriendly as possible, and see what happens.