The CDC and AAP recommend the use of products that contain 30% or lower concentration of DEET in children older than 2 months of age. Just make sure it is applied correctly (avoiding the eyes and not covering areas that have been sprayed with another layer of clothing). The best way to avoid tick bites is to wear long pants that are tucked into socks; apply DEET-based repellents to exposed skin and to pants below the knees and the tops of socks. It is recommended to use products with at least a 20% DEET concentration; the higher the concentration, the longer the duration of protection. For instance, a product with 7-10% DEET will last about 90 minutes, whereas 25% will last up to 8 hours. Some products marketed for families only contain 7% DEET; this will increase the likelihood of being bitten first as compared to someone who is wearing a higher concentration of DEET. Keep an eye on the child who is wearing the lower concentration.
Avoid sunscreen/insect repellent combinations. It is ok to reapply sunscreen, but bug spray is only reapplied if the insects start to truly bother you after a few hours. So, it is best to buy separate products. More is not better. It is also recommended that sunscreen be applied first, followed by the bug spray, otherwise the sunscreen will be less effective if applied at the same time.
Be sure to perform "tick checks" (don't forget the armpits, groin, and neck areas) as soon as you return home from your outings (even if it's just your own backyard) and wash the areas that have been sprayed with soap and water.
Disclaimer: I have no affiliations with Vertellus. I just really liked their information about DEET.