Question: I haven't looked at the Grand Canyon on a sectional yet, but I know that it is definitely big enough (by far) so that someone could fly a plane under the level of the rim and stil meet all of clearance requirements for VFR flight. 

Is it legal, and safe to take an airplane into the canyon?

Answer: 
It is generally not legal to fly *into* the Grand Canyon - there is a SFRA (Special Flight Rules Area) over much of the canyon which is designed to keep General Aviation traffic from annoying tourists (and away from the commercial air tour operators). For both GA and commercial tours there are altitude restrictions which keep flights well above the canyon rim.  

([You can download a chart of the Grand Canyon SFRA from the FAA](https://www.faa.gov/air_traffic/flight_info/aeronav/digital_products/vfr/))

Were it legal it probably would not be the safest idea anyway: Inside the canyon there are strong, often unpredictable winds. The conditions would be similar to flying a mountain pass, but with substantially less clearance between your wings and the canyon wall in some places.
Safe maybe, but not legal. Not anymore - the helicopter tours did fly into the canyon years ago, but it became illegal in 1987, prompted by a mid-air collision in 1986. [A Special Federal Aviation Regulations (SFAR)](https://www.faa.gov/about/office_org/headquarters_offices/arc/programs/grand_canyon_overflights/documentation/StatutoryRegulatoryandLitigationBackground.pdf) was adopted which regulates traffic below 14,500 ft MSL. This height limit can be raised seasonally to 18.000 ft, and only commercial tour operators are allowed to fly below. For GA traffic three small north-south lanes allow crossing at 10,500 (southbound) rsp. 11,500 ft (northbound). Now even the number of fixed wing and helicopter tour flights is restricted to reduce noise.