I want to get started with coyote hunting this winter.  I'm on a budget so mouth calls only- besides, I like the idea of the challenge of them.

I'm primarily caught up with what sounds I want to be able to make.  I assume the list is as follows:

cottontail distress
jackrabbit distress
squeaker
howl/bark/yip call

Is this a good list to start with? Do I need both a cottontail and a jackrabbit? Both species live in the areas I plan to hunt, but I'm not sure how different they sound.  Are there any calls I'm missing that are must-haves?

I use pretty much all the ones you list.

I have several variants of each, like a weems and Circe Jack rabbit call. They sound different enough.

You should be aware of Bobcats an mt lions. They're prevalent in Az

I've been predator hunting since the mid-80's and I still use an old Circe call some of the time. I think hand calls sound better than electronic.

I read an article by the great Gerry Blair many years ago. It said " do you want to be a predator hunter or do you want to be a DJ"?

A closed reed rabbit in distress call is all you need...... good luck.

So don't bother with other calls?

I have a cottontail call right now I'm practicing with, but I'm not sure what else I need.  I don't want to waste my time trying to hunt without the right equipment to be successful.

That's all you need. Pick up a jackrabbit if you see one cheap and/or you want a few different sounding reeds, but it's not needed. What you have has been calling in coyotes for decades..... good luck.

I've always had good luck with hurt/distressed woodpecker call.

I use a lot of coyote vocals and rabbit/rodent distress sounds. I'd personally get a versatile open reed call and a squeaker at a minimum.
Where in Arizona are you going to be?

Tucson area.  When I get back I need to do some scouting, but I think I'll try some of the state trust land off of 77 or 79 north of town.