While I like LBS bikes if you can afford it, there are some decent budget buys at some of the big stores. Pacific Cycle (makes Schwinn, Mongoose, GT, etc.), are available at Dicks and other sporting goods stores, and even Target. In this class of bikes, much of the basic components are the same as what Trek & Giant offers. Look for Shimano, SRAM, Sun. I would avoid Falcon (sometimes marked FIS), but that is just an opinion.
My daughters first gear bike at age 7 was a Pacific, and it was virtually parts interchangeable with a shop bike. I bought it used at a yard sale, and have since passed it on to a friend as my second daughter is several years behind. The big issue is assembly and maintenance. If you are not skilled, go to your LBS.
Given the terrain where we live, getting into gears as early as possible was important for us. She took to them quickly and was a much more enthusiastic rider as a result. One suggestion, however. Lots of supervision early on when your child makes the transition from coaster brake to hand brake. My daughter panicked and had a crash a week or two into it when she backpedaled rather than hand-braked. It is a hard habit for some kids to break.