In general, you will be considered a resident starting on the date you effectively reside in France. See How to handle moving between countries with different tax year dates? So no reason to worry about the first half of 2015, let alone 2014.
If you have been a resident in a given year (even for part of the year), you will have to file a tax return but not necessarily pay anything. Whether you are liable for income tax depends on the nature, source, and level of your income:
- If you have a job in France, you will have to pay French income tax following the same rules as anybody else. France does not have any pay-as-you-earn (taxation ‘at the source’) system so the first time around you will need to use a paper form and pay all your taxes in the following year (for income received in 2015, it will be in the course of 2016). After that (starting in 2017 for you), you will pay a provisional tax at the end of each trimester or (if you ask for it) at the end of each month, with the final amount determined in the summer based on your tax return.
- If you still receive regular wages from a private-sector employer (e.g. cross-border worker) or an old-age pension from Luxembourg, this income will be taxed in Luxembourg. You do have to report them to the French tax office and it can have consequences on the way other incomes are taxed but your Luxembourg-sourced wages are exempt from income tax in France. There are also some complications if you want to use the tax discounts that are available in Luxembourg.
- Other types of income or wages from other countries than Luxembourg (and Belgium or the canton of Geneva in Switzerland) are in principle taxable in France.
- In France, unlike Luxembourg, the marginal tax rate for the first bracket (up to about €10000) is 0%. So if you have a very small income, it's also possible not to owe any income tax, even without any special exemption.
That's the general case, assuming you would actually be considered a resident of France and work in either France or Luxembourg. But if you would be staying in or receiving income from some other countries, it's possible the relevant tax treaties would provide different rules.
The rules for other taxes are also different. For example, the taxe d'habitation (which tenants have to pay) is paid for the whole year by the person who enjoys the use of a dwelling on the 1st of January. So you won't have to pay that for 2015.