I would urge you not to get the cheaper grills, and instead spring for a basic Weber. You likely will have to replace the cheap grill every couple of years or so while the Weber will last. Another suggestion would be to get the Weber kettle and a chimney starter. You can get the kettle for less than $80, and even a 3 year old can get a decent fire going with a chimney starter.
The basic Weber grill, I think, is the Silver Genesis A, which will run you about $400. For $300, you can get a Weber kettle that runs on charcoal but has a gas ignition system. Again, I'll reiterate that if you want to spend around $300, I think you're best off with getting a kettle and a chimney starter and some starter cubes and going from there. My gas grill takes 10-15 minutes to get hot enough to sear steaks, and my charcoal, using a chimney starter, only takes about 5-10 minutes longer.