We just moved FROM Arizona to Pittsburgh. I know how you feel -- my daughter's 2nd birthday is in a few weeks and I'm feeling the same way.
I don't know which part of the state you're in, but if you're in either the Phoenix or Tucson area, the weather must still be pretty warm! Both cities have free "sprinkler parks." (I don't know about Flagstaff/up north...)
They're a lot of fun, and -- with proper supervision, of course -- totally safe for toddlers. I'm sure both your son and his friend would have a great time playing there, and you could take along a small picnic lunch, presents, balloons and a birthday cake or cupcakes.
It could be a nice, easy, low-cost way to both celebrate the occasion and to get to know your new city. Plus, there will be lots of other kids for your son to interact with -- you could even invite some of them to come over to the picnic area for cake, if you want. And the photo opportunities are excellent! Kids playing in sprinklers... what could be cuter?!?
You can look up the sprinkler park locations on either city's Parks & Rec website.
Good luck! Arizona is great -- I'm sure you'll come to love it :)