Although I won't go into great detail about the park, I did want to touch on some shorter hikes and places the the whole family can enjoy, even the furry members.
Both of these are shorter hikes, but will still allow you to catch a glimpse of Delicate Arch. If you visit in the early spring or later in the fall, while it is still cool, you can venture up to the Upper Viewpoint, which is only about 40-minute round trip. Obviously be mindful of your pet in the car and leave the windows open to allow some airflow. Before or after your trek, you can stop have a picnic in the car and give your dog a chance to walk around the parking area.
Although you won't be able to take the full 3-mile hike to Delicate Arch if travelling with pets, you can still make the shorter half-mile hike to the Wolfe Ranch and the Ute Indian Petroglyphs. Both were well worth the walk, and the Petroglyphs were stunning. You can also stop here for a car picnic, but the parking lot always seems to be full, even when we visited in February and March, so I would recommend the Upper and Lower Delicate Arch Viewpoint parking lot.
Another short 30-minute hike is Double Arch. What an awe-inspiring view standing underneath these two arches. It is almost like standing in an open cathedral. If you are daring enough, you can climb up and site right under one of arches, but the drop on the other side is considerable, so we passed on that experience.