I understand that large mountain ranges like the Rockies are created by the tectonic plates moving. I live in Arizona and there are a lot of mountains here but from what I understand, there are no plate faults here. How are the small mountain ranges are created?
There are no plate faults in Arizona (actually "plate boundary" is the term that we use - "fault" already has a different meaning, so it could be confusing), but there is still extension of the continental crust. The southern half of Arizona has part of the Basin & Range province, which stretches from Oregon to the south of Mexico.
If you imagine holding a peice of slightly brittle rubber and having someone push up from beneath it, you get cracks in the rubber where it stretches. This is happening to the North American continent. The 'ranges' in "Basin & Range" are the ridges in the rubber, the 'basins' are the cracks. (Exactly why/how this part of North America was pulled apart/pushed up is a subject of intense debate among geologists.)
You can intuitively see how these ranges would not be as high as the Rockies, which formed from a plate of oceanic crust (denser, heavier, thinner) being pulled/pushed underneath the contnental crust (lighter, fluffier, much thicker), causing continental rock to A) bunch up, and B) melt as water is added to it, leading to magma rising and bulging upward into volcanic chains. (Another factor controlling mountain height is erosion, so the Rockies, which are 80 million years old, are lower than the Andes, which are formed a similar way, but which are still being pushed upwards.)
The Himalayas are different again - instead of oceanic crust meeting contnental crust, this was contnental crust meeting more contnental crust, so instead of one side going underneath another, neither side will sink, they're both too light, so they cram into each other creating much higher mountains than either the Rockies or the Andes.
Just doing some quick googling, it sounds like Arizona also has both volcanic feature and eroded remnants of volcanic features. These kinds of mountains/hills are formed via a different method again, and their heights are controlled by their own method of formation. Volcanoes can vary a lot in height. They can be very small, like the volcanoes in south-east Australia where I live, or very large, like the volcanoes in the Andes mountain range, or they can be large, but broad and flat, like Hawai'i.
Erosion can carve out softer rock and leave behind more resistant stuff, and this is how some smaller hills/mountains are formed (like Uluru in central Australia). I found this online which might lead you in some interesting directions: Geologic History of Arizona By Jan C. Rasmussen