Where can I take my 5-year-old to play amongst other children?

Our family has recently been broken apart and I am looking to get my child engaged with as much social interactions as possible, before he starts his first year of school this Fall. We've done indoor play places, outdoor parks (seems so limited), and have tried setting play dates with extended family (not the best option).

We're on a budget, so having to pay to play every time is beyond us. Are there groups of single dads, or parents in general that meet to allow their kids to play with other children? I want to have healthy experiences for both my son and I, but without any emphasis on the "single" father aspect. Would love to meet people who are in similar situations and are focused on helping their child develop healthy relationships (I may be overcompensating due to the unhealthy relationship we're experiencing in our own family).

Honestly, if you have any great ideas in Houston beyond "playtime," that'd be great. Hiking areas, places to have picnics, sightseeing, or a special spot that we can take in spectacular views of this wonderful city would be great. I just want to do things with my son that I alone don't have the capacity of thinking about, humor me.

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EDIT: Thank you for all the wonderful suggestions. I'll be looking into each of them on days we have the availability to follow through.