re: I have 4 months in Scottsdale, AZ. Must do’s before I leave.

Hike of the west fork trail of oak creek canyon on Sedona.
Antelope Canyon and Horseshoe bend up near Page.
Monument valley up on the utah border.
Chiricahua national monument is a great hiking area.
Visiting the old mining town of Jerome and doing some wine tasting in the wineries around the town of Cottonwood is a nice day trip.
Tombstone and the old mining town of bisbee are options in the southeast corner.
As someone else mentioned the Heard museum is neat and depending on what time of year you visit the botanical gardens can either be a solid visit or can't miss.
The area around the white mountains is gorgeous and has lots of camping and outdoor stuff to do.
Since you are there for 4 months maybe check out Tortilla Flats and go for a paddle boat ride as well.

La Grande Orange and Il Bosco have great pizza (there are a crapload of great pizza places in the phoenix area, those are just a couple of my favs in Scottsdale).As mentioned PV burger company is a great burger joint with unique burgers. Persian room, the green corner, and Eden's grill are good greek/middle eastern. Definitely go to one of the Cornish Pasty Company restaurants at some point, they are great places to grab a beer and get some unique and very tasty food. Habanero's is solid mexican for lunch, Los reyes de torta down on 7th st is an awesome hole the wall Mexican place. I'm a fan of Otro Cafe downtown for Mexican as well. If you want some good breakfast Butterfield's pancake house is great.

ETA: Kartchners caverns is also great if you like caves.