In reply to verygneiss:
Are you looking for sport or trad or minimally bolted destinations?
And are you looking for single pitch, multi pitch or big wall type routes?
Do you have a preference for rock type? (Most popular EU destinations are limestone or granite but there is sandstone and gneiss to be found)
Off the top of my head, popular places people go in the summer to climb in Europe are:
Ceuse - 45 min walk in, single pitch sport.
Dolomites - 5 min to 3 hours walk in, multi pitch to big wall length. Mainly trad. Buses run during summer but a car is useful.
Chamonix - 5 min to 6 hours walk in, multi pitch to big wall length. Mix of bolts and trad. Lots of lifts that cut hours off the walk in but you do need to pay quite a bit for a pass.
Lofoten - lots of lovely pictures, granite, don't know much more about it. Except that with 24 hours of sunshine, you can never get benighted and if it rains all day, you can just snooze and then climb all "night" when it stops.
Ecrin - don't know much about this. More granite mountains like Chamonix but lower.
You can get to all these places with a plane and public transport (a couple places might need a thumb for the final stretch from a town to the base for climbing.
Give a bit more information on what type of climbing you are looking for and you will get more useful input I think.