The first idea is one that you've probably had already, the fan parks. The fan parks are usually excellent at promoting what football is all about: bringing people together. You'll be surrounded by other football lovers and have the chance to mix with supporters from other nations where you'll predictably talk about football. Its a safe place to drink and you're unlikely to bother any neighbours with the noise from what will most probably be a party atmosphere.
You can, however, have a change of pace from the football if you so wish though. Our first tip would be to escape the vicinity of any football ground in use throughout the tournament, fairly obvious right? What is key to know before you go though is where you will have to go and where you will want to go. For example, if you have tickets for two games Bordeaux you will want to plan ahead for somewhere you can get to (and get back from) quite easily.
Taking a day trip (or more) in a nearby city is probably your best bet to get some peace and quiet, or to get some decent sightseeing done away from the football crowds. You won't find yourself short of options either.
If you find yourself in the south-west, Montpellier will provide a contemporary break from the beautiful game and is within reach for a short notice excursion. An excellent city to sample some fabulous french cuisine, be sure to check out Comedie and Perou (The Royal Square) while you're here and you may as well sample some quality wine as well.
Of course, there's no real need to even stay in France, as is the convenience of so many neighbouring countries, meaning both Switzerland and Belgium (and even a trip to Barcelona) could be viable depending on your timeframe. So getting away from it all shouldn't be a bother at all.
We would also recommend heading to the outskirts of the host cities, again away from the major footfall. It is here where you'll stand a real chance of meeting and bonding with the locals and you may even be rewarded by being pointed to some little-known gems of the cities, be it a quiet pub or the best place to grab a bite to eat!
Finally, carry the knowledge that prices for travel, hotels etc. during this time will be hiked up. So book early and/or take alternative measures. Camping, for example, could be an excellent way for you to justify spending more on excursions by saving on hotel costs. Do your research and use a little nous and you'll be coming home a winner, regardless of the result.