I took the trip at the end of June. The weather can be very hot, but at the beginning the river water is very cold! Most people wore waterproofs to keep most of the water off in the rapids, but if you don't mind getting cold for a while until the sun dries you off then they are not essential for that time of year.
The rapids on the first couple of days are relatively small, and let you get used to how and where on the boat you want to ride them. Right on the front of these boats is the wettest ride on the river! The rapids get (much) bigger - Spectre, Doris, Lava Falls and all their brothers and sisters are waiting for you!
Most of the people on your trip will be American - I was the only Brit among 27 people. In summer it is normal for up to half the passengers to be kids/teenagers, which surprised me - they were all extremely well behaved on my trip, but I doubt if they fully appreciated the experience at that age. If you prefer to avoid youngsters, check the American school holidays.
Pack light! Quick drying clothes, light shoes/sandals for on the boat (the sand can be very hot when you get off, as can the surfaces on the boat when you get back on) and walking shoes for the hikes exploring side canyons etc.
If you like to take photos, take a waterproof camera, even if it is secondary to your main camera. There are many times when it would be good to take photos in or near rapids. And lots of batteries!
This is wild camping done the American way. Camp beds, mats, groundsheets, sleeping bags provided, portable toilet, enormous amounts of great food.
The Grand Canyon is incredibly clean and there is absolutely no evidence that on any night in the summer up to 1000 people are camped along the river. There are rules to follow to keep it that way, but they are no hardship and are the right thing to do. Please help to keep it unspoiled for future travellers to enjoy.