For France I have two special favorites. Schweppes Agrum (citrus) has already been mentioned. I love it, but have only seen it in France. I looked in supermarkets in Spain and Switzerland, but while they have Schweppes Lemon, they don't have Schweppes Agrum. And at a cafe in France (not usually at a restaurant), you can get a citron pressé. This is fresh squeezed lemon juice, water, and sugar, all served separately; you mix them to taste. This is what you should order if you want "lemonade"; as said above, a limonade is something different.
Many countries have their own local specialties, some more palatable than others. At supermarkets they're cheap, so it's fun to try an unfamiliar beverage. One bottle of Irn-Bru in Scotland was sufficient for me, but perhaps you'll like it more.
If, like me, you love carbonated beverages, note that can be orders of magnitude cheaper in supermarkets than elsewhere. If you're buying a bottle of soda at a cafe, or even at a snack cart, you just have to accept that it will be several times the supermarket price.
Whether you can get free tap water when you are not ordering other beverages varies from place to place. In France it's common; just ask for un carafe d'eau, s'il vous plait (the tap water does often come in a fancy carafe). You'll see that most other diners in the restaurant will have one on their table too. In some other places, they will only give you tap water if you're ordering another beverage; if you aren't, you will be expected to buy bottled water. The cost and size of this varies; in Italy it's usually 0.5 liter or 1 liter and is cheap, while in the Netherlands it was 0.33 or 0.5 liter and cost more (restaurants in the Netherlands rarely had 1 liter bottles of water available).
And it hasn't come up in this thread yet, but it will come up sooner or later, so I'll be the first to say it. In no place in Europe, and yes that includes France and Italy, is there an expectation that you will be drinking alcohol. Your server may ask if you want wine, but they're just being thorough (similar to asking if you want dessert); feel free to say "no," and that will be the end of it. But I can only recall that question a few times. The rest of the time, you will be asked what you want to drink, and the server will bring whatever you ask for; that's the end of it. The various rumors about how you will be sneered at if you don't have wine with every meal are just that. I often get a Coke Zero with meals, and I've never had so much as a raised eyebrow at the request. Don't worry about having to come up with any explanations for why you're not having alcohol; no one will be bothered.