First, AZ law states that anyone 18 or older, who is legally able to own a gun, may carry a gun openly. Open Carry means that all or part of the holster is visible and clearly identifiable as a holster. Just wear a gun and don't cover it up. Carry it loaded (ain't no good otherwise) with 'one in the pipe'. In your car, the gun must be either a) in plain sight (dash, passenger seat, etc); b) holstered; c) in a glove box, center console, or map pocket; d) in luggage.
If you are 21 or older, you may carry discretely if you chose (partially concealed, totally concealed, whatever) without a permit. In a car, you may conceal anyway you choose (under seat, etc).
You can carry just about everywhere (no preemption by local law). No such thing as an 'e-check' here. Stay out of federal facilities [post offices, and national park or national forest offices], stay out of schools (no 1000' rule, guns on school property must be unloaded and out of sight in car), stay out of voting places, and nuclear or hydroelectric plants.
Private businesses will post signs ("no firearms" or some such) if they don't want guns inside. Those signs have no weight (just like 'no shoes=no service') but you could be arrested/cited for trespassing if you refuse to leave when asked. Generally, such signage means 'conceal carry' to me.
Any place that serves alcohol (bars or many restaurants) are no-go for open carry. Unless posted, it is ok to conceal carry with a concealed weapons permit-but don' drink. If such a place is posted 'no firearms' [official sign = generic semi-auto pistol in red circle with red line] it is generally at the door or at the liquor license. Such places are illegal to carry a gun by law.
Sedona is fine to carry. AZ is gun friendly, and Yavapai county is very gun friendly. It doesn't get any more liberal than Sedona, and it is not a problem there (individual businesses may vary).
Grand Canyon is OK to carry (just stay out of ALL buildings, as they are fed owned) because it follows state law.