I spent a week in Valencia last summer in August and absolutely loved it. It had everything - historical city center with narrow cobble stone streets opening on to wide plazas, beautiful palaces and cathedrals, expansive beaches, and excellent parks and outdoor spaces. We found it very bikeable, and it had excellent public transportation. There were lots of museums and activities for kids, as well, and a distinctive culture. We stayed in the old city, and it wasn't too hot in August, because there was a steady breeze. I imagine being closer to the beach would be even cooler. Plus, you are close to Barcelona should you want to train in for a couple of days. A quick airbnb perusal indicated that you would be able to find a 2 bedroom apartment within your price range.