Before you decide on timber size and connection you need to design the overall structure. 6x6 will likely be too small for mortise and tenon, and likely too small for developing tension joinery. Maybe for steel plates.

You can use a drill and chisels. I wouldn't go right angle, I'd pick up one of the Milwaukee drills. I've used and older model of this:

http://www.milwaukeetool.com/tools/drill...-350-rpm/1854-1

You should really get an engineer or designer involved. A picnic shelter is an expensive way to build a kite.