This is a fun mountain to climb and there are several trails to choose from (but I believe they are closing Echo Canyon for several months, check the website). Love bringing my dog here.
Camelback Mountain isn't a difficult climb by any means, but you will definitely need to bring water for your and your four legged friend. There are many vista points where you can stop and catch your breath if you need to, but be aware that the trails here are pretty crowded compared to those you would find in Sedona or other places outside of the valley.
The first 45 minutes or so is a pretty gradual descent up to between the camel's humps, and then the last bit of the hike will require a little bit of climbing before you summit. I recommend bringing a Cliff bar or something with you for the top to reward yourself for a job well done. The views are breathtaking--360 degree of the entire valley and beyond.
When you come, you will most likely park in the Echo Canyon lot or along a street near the neighborhood of the other trailhead. Be sure to stretch a bit beforehand and get ready for the best viewpoint in the valley!