Gazebo/Small Backyard?

Hi all, ok so I have a small backyard, big enough though to fit a 10 x 10 gazebo....I would say my backyard is 30 x 30 part gr and part concrete (equal). Anyway, I want to be able to be sheltered form the sun, have my kids protected from bugs and all that good stuff. My question is are there any other ideas besides a gazebo? Am I crazy putting a gazebo in a small yard? Do I put it on the concrete or gr? And if gr, how do you mow? First time home owner, pardon the greeness:-0)