We got the All Terrain Carrier Wagon just before my son out grew his stroller (about 2 yrs old). He'd want to walk, but he'd get tired quickly, and the wagon was great for pulling him around (and he could lay down and take a nap when he wanted. We'd put in a few beach towels, and our water bottles (and the wagon fit just right into our mini van) and we went to county fairs, Renaissance festivals, and parks.
http://www.radioflyer.com/wagons/all-terrain-cargo-wagon....
He's out grown it a long time ago, but it's great as a garden cart for carrying my tools and gadgets about the yard.
There are some other great options wagon wise right now.
A garden wagon is about the same size, comparable in price, and can carry a lot more weight. With an indoor/outdoor cushion and/or a foam mat, and a towel or two, it will be very comfortable and you can hose it down to clean it and you'll use it for years to come.
http://www.cartsandwagons.com/wagons/lawn-&-garden/gr...
There are some clamp on umbrellas that will work on anything (we had one but didn't use it much, it was a bit of a bother to use). Seat belts were not a concern for us as our boy just climbed in and out. By the end of the day he was usually sleeping and a lot of people commented how comfortable he looked.