I hike the Superstitions pretty often, love it up there. My favorite spot is on the northwest side. If you take RT 88, go about a mile or so past the entrance for Lost Dutchman State park, to first water rd/trailhead. It turns into a dirt road..There are several places to park and hike, but if you go 3/4 to a mile down the road there will be another dirt road that branches off to the right and will take you towards the mountain.
Take that all the way down, there will be a fence with an opening you can walk through for hikers. From there it depends how far you want to go, but it's gorgeous.
Another good spot that is less known on the Supersitions is "hieroglyph canyon." If you do a google search for it, you will find directions on how to get there. You have to go down Kings Ranch Rd in Gold Canyon, but it's been a while and theres a lot of zig zagging to get back there so I would definitely look up the directions.
Here's a few pictures from there, if you go after it has rained there will be running water which is just an added bonus.
This is near the beginning of the trailhead.
Some petroglyphs
Peralta is also a good hike as you seem to be familiar with. Once you get past weavers needle the traffic gets very light..Not many people go past Weavers Needle.
If you have anymore questions feel free to ask, I love the Superstitions!