Encanto Park is a pretty good city-center (almost) greenspace for Phoenix. I use the term greenspace loosely, since all the grass was dead when I visited.
Besides the groundskeeping issue, there are a number of nice options here. There's a few paths for strolling, a small lake/lagoon for boat riding, as well as an adjacent golf course and tennis courts. In addition, there's a small amusement park with a train ride for kiddies and even a carousel.
Although I didn't jump on the carousel horses or other rides, the amusement park seemed entertaining (in the way that an adult can guess what might seem interesting to a kid). I would also mention that they have a small old school arcade at the side of the park. I almost kept on walking, until I saw that one of the games was PunchOut. Since I spent a small fortune playing this in younger days, I stopped to watch someone start playing; only to get quickly knocked out by Glass Joe (the first boxer). I fished in my pocket for a few quarters and smiled. "Hey kid, let me show you a trick or two for this...."