Classic areas out West with many traditional multipitch routes and not too hot:
devils tower - chase the shade - share a campsite to save money
RMNP - free once you pack in
Wind rivers - deep lake, cirque areas for large number of routes in one place at many grades - free
Elephants perch in Sawtooths - free except for the boat ride
Squamish - share a site
entire east side of Sierras - free
Tuolomne- share a site
Needles - chase the shade, free
city of rocks - chase the shade but not multipitch in general - better to just chill out and re- energize for the mountains. Free if you go outside to camp or share a site right in the park
Canmore area in Alberta - depends on where you go for camp costs
Bugaboos- hut fees
Tetons- free once you hike in
Be sure to check snow depth prior to planning trips as many of these mountain areas are snowed in until early to mid July. It just started to really snow in the southern areas which might lead to a June that is hard to get in to higher altitudes. With the exception of a few of these you can get all the climbing beta and approach info off this web site as you will probably be doing one big route a day. For places like the city and needles and Tuolomne and Squamish a guidebook will save you a lot of hassle when searching for routes as you can do many per day.
During the summer it is way too hot in Southern Utah and Red rocks and J tree or Yosemite to justify a road trip there in the summer when you could as easily go to these other special places further north or higher altitude - some which have very short climbing seasons. Realize that most of the trad routes in a few of these mountain areas are old school and the ratings don’t really correlate with gym ratings so choose somewhere with a broad range of grades near and a bit below your comfortable leading limit so you don’t go there and get skunked after all the effort required to hike in and set up a camp. If you want single pitch or sport climbs there are many areas between these places for more casual or shorter days.
Have fun! A lifetime of memories await you.