Awesome! I love living here, I've been in Tempe since 2004. There are definitely some things that the area needs work on, but things are changing for the better.
Drink a lot of water, all day everyday. The water here tastes bad if you are coming from someplace with a well, or with delicious water. You'll get used to it. Our water comes from three places, the Salt and Verde Rivers, and the Colorado River. It has to travel hundreds of miles through pipes to get to us, so it tastes metallic. We have a filter on our fridge, and I have a huge mint plant that I add a few leaves from. I know this sounds terrible, but the weather is absolutely beautiful for 8 months of every year, so it's a trade-off.
In addition to the other parks mentioned, Tempe Beach Park is fun for kids, there's kiddie splash pools with fountains and there are always a lot of kids there with their parents in the summer. There are also a lot of good bike/running trails there that go around Tempe Town Lake. Chaparral Park in Scottsdale is nice, and great for dogs. If you live in a historic Tempe neighborhood there are a lot parks, Mitchell park and Daley park are good for kids and dogs. You can rent the ramadas at Daley park and get a drinking permit and have parties there. Cesar Chavez Park in Phoenix is nice, and Civic Space Park has outside yoga on Saturday. Also, Papago Park. Papago park is a tri-city effort encompassing the area where Phoenix, Scottsdale, and Tempe meet. It is right by the zoo and the desert botanical garden. It is great for hikes, and full of desert fauna/flora.
As the previous poster said, Green and Pomegranate cafe are awesome. But Desert Roots Kitchen is also great. It's a cold deli located off of Mill Ave in Tempe, open 9am - 330pm. It's vegan, and the owner is fantastic (and the reason I went vegan.) La Bocca, C.A.S.A, Thai basil, Cornish Pasty, the Breadfruit, and Cowboy Ciao, Udupi, Tamale Store, Pita Jungle, Juba, Z Pizza, Haji Baba, and Phoenicia Cafe also have vegan options, you might just have to ask the server what they recommend. And I haven't been there yet, but I think D'lish (healthy fast food drive thru) has a black bean quinoa burger. It might be vegan. Usually can get something everywhere, but it will probably be hummus, guac or salsa and chips, edamame, or maybe a pretzel (ask!). Either way, you need to go to Four Peaks Brewery. It is so so so good.
For grocery shopping there are Whole Foods, Trader Joe's, Safeway, Sprouts, Fry's, Walmart, Target, and Fresh n' Easy peppered throughout the valley.
I usually go to a farmer's market (the kinds with tents) to get my produce and sometimes bread. There are about 8 or 9 throughout the valley on different days, but the biggest and best (in my opinion) is the Phoenix Public Market. It is open Wednesday night from 4pm-8pm, and Saturday morning from 8am - 12pm. It is awesome. Horny Toad Farms is my favorite (they also have a shop at Roosevelt and 3rd ave in phx), but I also like One Windmill Farms (great beans, wear green shirts), Golo Family Farms, and the IRC New Roots program. Sunizona farms also has a co-op you can participate in (they're a vegan farm) and pick up from around the valley. So I get my veggies from a farm, and then go to Whole Foods for bulk beans and grains and get snacks either there or from Sprouts, or Safeway.
I also go to the Japanese market on Priest and University for Soba noodles and bok choy, and right next to it is a place called the food ministry where they have "old food" it's not bad, but it might be close to expiration. Well you can get some sweet deals there.
For shopping, I am a poor student so I can't say much. But I get most of my clothes from Sunset Clothing Exchange, which is by far the best used clothing store I have ever found (They have a three dollar rack, but also sell brand new Paige jeans for $60). In addition, there is an antique store on Scottsdale road called the Antique Trove. It rocks. The goodwill near old town is just a furniture store, so you should check that out, too. Also consider heading out to the Goodwill in Paradise Valley or to one of the many estate sales that happen during garage sale season.
Tempe Yarn and Fiber has knitting classes. There are all kinds of sporting/activity classes offered through the citites of Tempe and Scottsdale and they're about 8 weeks long. It's a good way to meet people and they are not super expensive. Mesa Center for the Arts has art classes you can sign up for. Scottsdale CC and Mesa CC are local community colleges and have tons of classes and it's pretty inexpensive. Around $80-$90 per credit. But yoga and dance classes are one credit, and are three months long, three days a week. Also, rythmic expressions has dance/yoga classes for $7 each. There are a lot swanky studios (ballet barre, balley body, pole dancing, pilates, bikram -type places) around, I just can't afford any of them.
While you're here you should visit Tucson, Sedona, and Flagstaff. Sedona and Flagstaff are great if the heat starts to get overwhelming. Also, Vegas, Mexico, and San Diego are about 5 hours by car or 45 minutes by plane.
If you like live music the Crescent Ballroom is no doubt, the best place for live music in the area. The Sail Inn and the Yucca Tap room also have some good shows. As does The Rhythm Room.
As for critters, it seems that scorpions and black widows like to congregate. I have never seen any in my own house, but a friend of mine who lived a mile away would have them in her condo all the time. Scorpions will glow under a blacklight. Also, under AZ law, if you find a bark scorpion you have to kill it and not let it back into the wild. Other than that, there are some huge bugs but they do no harm.
Wear sunscreen, but get 15 minutes of sun per day without it.
Take your kid to the children's / science museums.
Learn to ride the light rail. It is fun. The orbit buses (in Tempe) are free and circulate around Tempe neighborhoods. They take you to the light rail station, Mill Ave, the public library, and many parks. Also, the regular buses can be frustrating at first but they are air conditioned and easy to get used to.
And drink more water.