You will need a car. Tuba City is on US 160, the main east–west road across the Navajo Nation. From Flagstaff it is 130 km via US 89 north and US 160 east — about an hour and a half. From the south rim of the Grand Canyon (Desert View) it is about 110 km, also via US 89 north. From Page and Lake Powell it is around 120 km on US 89 south.
There is no scheduled bus service and no passenger rail. The nearest airport with commercial flights is Flagstaff Pulliam (FLG), with a few daily connections to Phoenix and Dallas; Phoenix Sky Harbor (PHX) and Las Vegas (LAS) are the larger gateways, both about a four-hour drive.
Note that the Navajo Nation observes Daylight Saving Time, but the surrounding state of Arizona does not — so for half the year the local time in Tuba City is one hour ahead of Flagstaff and the rest of Arizona. Plan accordingly.