Recently, my family and I traveled down to Lake Balaton, and ended up seeing some extra sites that are worth sharing. So, if you are looking for day trip ideas, I would suggest checking out the Laposa Birtok Winery and the Szigliget Castle.
From Papa, the Laposa Birtok is about a 1 hour and 10 minute drive. It has breath taking views of Lake Balaton and some very tasty wine. For more information, please visit their website at: http://www.bazaltbor.hu/en
Next, you can drive another 15 minutes, and see the Szigliget Castle. This is the ruins of the castle, but has activities and rooms to explore. It was fun to walk around and see the castle, while also giving you amazing views of Lake Balaton and the areas around the castle. For more information on this, please go to: http://www.szigligetivar.eu/