@ArizonaSquirrel
I've never lived out there like
@azbackpackr, but I have spent quite a bit of time out there.
The town of Alpine is the closest to the hikes that you mentioned. The setting is beautiful, but there aren't really any permanent cultural attractions, like an art district. However, they do have events throughout the summer on weekends, so if you check the
calendar you may be able to plan around something that interests your family.
That said, the town of Greer is a little more touristy, has a few more restaurants, and probably does a better job catering to vacationing families than Alpine or Springerville. That may suit your needs a little better. I took my family out there a couple summers ago. We went horseback riding and also spent a day doing the summer activities at Sunrise Ski Resort. Other than a general lack of organization at the ski resort, and the poor wifi at the cabin we stayed in (preteen kids), it was a pretty nice time.
In terms of family-friendly hikes: Thompson Trail is as good as it gets, in my opinion. Follows a quaint river the entire route, so you aren't committed to a certain distance before reaching a destination; it's just a nice, easy walk in a pretty setting, and you can turn around whenever you please. The West Baldy trail is also very nice, and similar in that you are following a river with minimal elevation change for the first couple miles. Very pretty, and you can turn around at any point without feeling robbed of a particular destination.
Aker Lake and Hannagan Meadow are also both very pretty and family friendly. Neither of them a hike, though. Aker Lake is a quiet, hidden gem if you want to kick back and enjoy the sunrise or sunset. Escudilla is nice, but may not exactly be family-friendly . It's a steep climb up and your reward is an incredibly burnt forest. It doesn't have the same mass appeal as the myriad of creekside hikes out there.
In terms of weather: Your best bet in July/August is to do your hiking in the morning. Afternoon can bring some pretty fierce and unexpected thunderstorms. Regardless - pack raingear no matter what.