Fly into Dublin in June, you should be able to get a better deal that way. Just keep an eye on travel cost from Dublin to Continental Europe.
If you are rushing around and prioritise (say you go and see Barcelona and Pamplona instead of Spain and Paris instead of France) you can plan to spent 10-14 days per country.
You can find out about train travel times by looking at http://reiseauskunft.bahn.de/bin/query.exe/en and typing in your destinations. Works for most of Europe.
I would like to hike and see the natural beauty as well as experience the culture
You can only do that if you travel on a slower pace. Culturewise it takes time to "get" a country, you don't start picking up the little things that are pieces of the puzzle until you have been there for about 2-3 weeks.
Natural beauty especially takes time. Did you know that one of the big long distance hiking paths of Europe passes through Pamplona? It is the St. James Way, an old pilgrims route on Roman roads that leads to Santiago de Compostela. Doing pilgrimage has recently become hip again, not just among young Catholics.
[ Edit: Edited on 21-Jan-2009, at 20:17 by t_maia ]