If you're breastfeeding then you don't have to worry about food. Your child will need something to sleep in, we've used everything from a car seat liner to a sleeping bag folded in half, but the best is probably one of those sleeping bag style jumps that have should straps.
A hat is always a good thing to bring one with a chin strap, as is a source of shade (tarp, beach umbrella, etc.) if there is no natural shade, but since your child is only 3-4 months old a blanket over a removable car seat is probably fine.
For carrying the child, I couldn't recommend anything better than an Ergo Baby, we first used ours when our son was 3 months old, and he slept the whole time he was carried. It feels great carrying them that way, and they are close and warm, although they look like a spatchcocked chicken when you peel them off your chest.
At that age you don't need much more. Lots of garbage bags to wrap up dirty diapers, it's worth learning how to knot off the bag after each diaper so that you can put about 5 diapers in a bag without the smell escaping (much).