Under EU law, you can indeed vote in the municipal and European elections. Beyond that, a few countries (Netherlands, Denmark, Malta, the UK) allow EU citizens to vote in some other local elections. British citizens can also vote in Ireland in some national elections (but not in referendums I think) and vice versa. France does not allow anything more than what is mandated by EU law so only municipal and European elections for EU citizens.
To vote you must register and since there is no general registration system in France, this is simply a separate process (inscription sur les listes électorales). It's done at the mairie (town hall) where you live. It can also be done by mail or online. You will need a proof of where you live (typically a utility bill).
One thing about France is that the deadline to register is long before the election. To vote in a given year you must in principle register before the 31st of December of the previous year (so typically 3 to 5 months before the elections as they mostly take place in the spring). If you would register now (or even back in January), you would only be added to the electoral roll in 2015.
There are however exceptions, for example if you had to move “for professional reasons” (that's the way official documents put it, I don't know if you would qualify or how you would go about proving that). In that case, you can still register until the day of the election. (I don't think there is any firm deadline but towns do need some time to process applications so in practice you should do it at least a few days before).
Incidentally, it's still relatively easy to become French (and was even easier yet in the past, e.g. under the 1793 constitution as anyone could become a citizen without condition after one year of residence). The traditional view is therefore that offering people citizenship (and not granting rights to foreigners) is the main route towards broader political participation.
