Maybe I'm more cautious than some parents - but I always have a first aid kit with me. My diaper bag has multivitamins and sunscreen and nail clippers. It also carries my son's Epi-Pen, since he has a peanut allergy. In my car, I keep a first aid kit with the basics - bandaids, burn cream, gauze, tape. This doesn't seem excessive to me in the least, but my husband jokingly teased me for my paranoia. Personally, I'd rather be safe than sorry.
Summertime means that kids spend more time outdoors and in the sun. Of course, this means they're most likely to get bug bites or sunburn - but that's not inevitable. If your kid is over six months, you can slather them down with baby sunscreen and bug spray with up to 10% DEET. As a side note: many parents prefer "natural" treatments for bug spray - particularly ones that include some form of essential oil. Now, there are studies that show that Lemon Eucalyptus is comparably effective (for a shorter time window) than more conventional sprays. But here's the catch - there are no scientifically valid studies on essential oil use on small children. So, use your own judgment before automatically choosing the "natural" option. Natural isn't always better.
I know this seems standard, but I don't just mean your regular old sticky kind. Sterile gauze, butterfly closures, and blister covers (or moleskin). All come in handy when you're out and about. At the very least, butterfly closures can keep skin together on the way to an Urgent Care for stitches. Hey - it happens!
In addition to your standard Neosporin/Triple/Bacitracin, I'd recommend adding a good burn cream. I personally add a stick of After Bite, but I only use it on my toddler. I include diaper rash cream on here, as well as good old Chapstick - both make good protectant layers on bruised or scraped skin. If you only want to carry one thing, lanolin is a good option. Fair warning: you don't want to cover a burn or sunburn with lanolin, as it will trap in the heat. Use aloe or burn cream instead. Cortisone creams are nice for any bug bites or itching from things like poision ivy or sumac.
Baby wipes are LIFE if you're a parent, and first aid kits are no exception. But beyond just basic cleaning, you want some wipes that do a little more antiseptic. Alcohol wipes or betadyne/iodine are good to have on hand to help clean up scrapes or open wounds. But honestly: baby wipes will do in a pinch. I also carry a pack of antibacterial wet wipes in the car - helps keep my hands clean after diaper changes or before treating a boo-boo.
Bug spray is basically the only way I survive summer. I am a universal donor, which is just another factor that attracts biting bugs to me - and boy, do they feast! My pediatrician gave me the okay for sunscreen and bug spray after six months. I recommend Blue Lizard products but anything with a 30+ SPF will do. Don't forget the aloe and anti-itch to treat those bites and burns that do happen.
Baby versions of standard medications - Tylenol or Motrin, Benadryl, gripe water, Orajel, Mucinex, Claritin - are a must. I also have a dropper and a spoon with measurements and a conversion chart based on weight. That's maybe a little overboard, but hey - it helps my anxiety.
More Serious Stuff
Remember when I said my kid has an Epi-Pen? That's in the "more serious stuff" category. Any life-saving medication or device - like insulin or an inhaler - is a must-have. Don't go anywhere without them! You never want to need them and not have them. Also: gloves, splints, or syringes. Don't forget those!
What else should I be packing in my first aid kit? Any specific brands you prefer or convenient travel sizes I should know about? Find me on Twitter @pi3sugarpi3.