I have a great suggestion for a great picnic place while you’re visiting the old city of Jaffa, it’s Midron Park.
So, after a nice walk along the old and magical streets of the old city of Jaffa, visit the Jaffa port (you can grab some tasty drinks and pastries at the LovEat over there, check the LovEat article) and walk to the south along the sea until you get to the first hill. The second hill has many great small trees which provide nice shadows when the sun is high and about. Seeing the sunset from this place is my favorite thing to do on a Thursday evening. At night you can come too, with a blanket and see (mostly) a clear sky with lots of stars.
You can find a kid’s park between the 2 hills – a great place to visit with kids and let them run around while you’re chilling under a tree :).
Pay attention – at the weekends (noon and night) and Jews/Muslim holidays this place becomes toooo crowded- I would not recommend you visit it then.