First up...for the little ones:
These are 'Trekkers' from LandsEnd. I love this for travel because these closed toes sandals protect kids feet very well. I also like that they come in a great array of colors for both boys and girls and NO LACES! No one wants to be stopping the fun to tie laces...especially not when your at any kind of park. Hopefully of the amusement or water variety! :)
There are quite a few companies who make variations of this shoe. My kids have also had the Jumping Beans brand available at Kohl's.
Now don't get me wrong gym or tennis shoes and socks are fine if you don't mind being hot and sweaty and stuffy on the foot all day. I also like the flip flops you can find around town for as little as 99 cents. They're great for the pool or splashing around but if you can find a Good closed toed sandal that protects the foot, let's them breath and is no hassle...Go for it!
For Mom and Dad:
The boat shoe.
These Sebago leather shoes are nice but obviously if you invest in leather...just fly in them. The boat shoe is great for flying in because it slips on and off easily and if you've broken them in enough they're extremely comfortable.
Again, there are tons of options when it comes to the boat shoe. There are designers such as Prada and Sebago. Brands like Sperry and Polo as well as companies who make this style available at every price point. I found an awesome pair at Target a few ago.
These leopard print shoes are available at SperryTopsider.com.
As for trekking across large parks, sporting around and the overall length of your stay I also recommend something similar to my kids pick. A shoe that you can walk, run and spend all day in comfortably. A shoe with the right support technology.