DIRECT TO GRANADA
Since Ryanair pulled out of Granada there are not many direct international flights to Granada.
There is still a direct flight from Paris to Granada with Vueling
Info here http://vueling.com/
If you go with Vueling to Granada via Barcelona there are plenty of possibilities you can even fly from Moscow, Edinburgh, Brussels, Milan etc.
A flight from Amsterdam via Barcelona would cost 104 euros.
There are indirect flights with Iberia via Madrid or Barcelona. IBERIA.com
FLY TO MALAGA THEN BUS TO GRANADA
If you want a very large range of possibilities and cheap flights it is best to fly to Malaga.
The best website I have found so far is SKYSCANNERSKYSCANNER.com
There are loads of flights from all over Europe.
For example a flight from Copenhagen would cost 65 euros.
The easiest way to Granada from Malaga is by bus.
There is a bus stop from within Malaga airport which takes you to the "estación de autobuses" (the bus station)
You can either buy a ticket at the station or buy one online at http://alsa.es
It takes about 90 minutes.
Once at the Granada bus station it takes about 18 minutes into the very centre of Granada by bus
More info herehttp://www.granadamap.com/bus/index.htm