June is an excellent season. But try to book the hotels as there will be many tourists. In general I can recommend May-August. There might be rain, but the weather is warm and sunny (the temperature can go up to 35C. But take also some warm clothes and a summer rain coat with you - just in case or when you are on top of a mountain or must walk through the rain.
I would recommend:
Geneva
Zermatt (Matterhorn)
Bernese Oberland
Lucerne
and then go further to Grisons (Graubünden) for
St. Moritz and Sils or the Unterengadin.
If you have time, you might want to include:
Zürich
Rhinefalls
Titlis via Engelberg (train from Lucerne)
There are also sight seeing trains:
http://www.mob.ch/GPS/main.asp
http://www.glacierexpress.ch/
The train system is excellent and you can reach all places quite well. For timetables:
www.sbb.ch
Good Luck on your journey through life and Switzerland