If you're traveling from sea level be aware of the sharp change in altitude while recreating in this area. You and your dog will be in the mountains and you need to take precautions. Be sure to give yourself and your pup a few days to adjust and take it slow when you do start hiking. Bring plenty of water on your hikes and be sure to take frequent breaks until you adjust.
Many of the trails you can take your dog on are in National Forest, so be prepared when you are headed out on hikes. Bring a map and know where you're going. Wear appropriate footwear and bring a day pack with plenty of water, snacks, and layers. Bring some extra for your dog too. It's also important to do a little research to determine where your dog can play off-leash and when you need to hook in. Following the rules will make sure that dogs are allowed on trails for many years.
If you're looking for a longer day out, head to the Duck Pass trailhead in the Inyo National Forest. This trail is steep with lots of climbing, but you'll be rewarded with stellar views and a multitude of lakes as you make your way up the Coldcreek watershed. Your dog will have plenty of opportunities to cool off in these lakes and you'll have time to take a rest before you climb up to the pass.