Rainbow Parking area is between McHugh and Falls Creek. Trail starts at north end of parking lot. You will walk along the Turnagain Trail for a while (maybe a mile) until you get to its highest point. At this point you will be on the crest of the ridge and you should see a faint trail leading up the ridge on your right. There are actually two of these trails -- either one gets you on the ridge. It is a little steep, especially at first -- so don't be shy about using the cottonwood trees for leverage! Once on the ridge, the trail is pretty well marked and at the top of the ridge you pretty much get cliffed out -- but the trail veers off to the right, along the base of some pretty steep walls. Be aware that there are a lot of game trails so it can be easy to get off the right trail if you aren't paying attention. Once you go a few hundred yards, the steep cliff gives way to a broad bowl, which is pretty much loose scree with a few outcrops of solid rock. I prefer walking on solid rock over scree (at least uphill...) so I kind of hopscotch from rock exposure to rock exposure. Stay towards the middle of the bowl and keep going until you gain the ridge at the head of the bowl. Just a little scrambling gets you there. From there, it is a delightful ridge walk (the peak is to your left) with a nice view of South Suicide and Turnagain Arm. You'll know you reached the summit when you can't go any higher!
A challenging alternative is to climb Rainbow, then backtrack along the ridge but instead of dropping back into the bowl, keep following the ridge to South Suicide. It gets a little steep going to the summit of South Suicide, so be prepared for some route picking. (I wouldn't advise this option in the winter!) Then you can return via Falls Creek, with a bit of a hike along the road back to your car.
It is my favorite "after work" hike because the trailhead is easy to get to, and the hike only takes a few hours round trip. It is about 3500' of elevation gain and about 5.5 miles round trip. What I like most about it is the variety -- you get a little serene woodsy walking in, some steep scrambling that can be as challenging as you want it to be, and my favorite, a good ridge walk for about a quarter mile.
Dogs can go on this hike as long as they are okay with a little exposure.
It can be a little windy up there so bring an extra dry shirt (if you are like me and sweat like a wildebeest, you'll need it!)
I have climbed Rainbow in every month of the year -- it is one of the first to be free of snow in the spring. During the winter, it is best advised to wait for a snow-free period and give the snow a chance to blow clear. Even in January it is possible to reach the summit with little to no snow travel.
Bears are known to frequent the area along the Turnagain Trail. It is not an especially brushy or concealed trail, but be aware of their presence.
Moose are also regular inhabitants and I would argue that they can pose just as much of a hazard. So just make a bit of noise and keep an eye on your dog.
Lastly, stay on the trail in the first part of the hike -- there is, I believe, private property that you go near.
A Chugach State Park parking permit is good at the trailhead. I have heard reports of cars getting broken into there, so don't leave any tempting treasures for the little rapscallions. I would recommend that you park near the entrance off of the Seward Hwy because the car will be most visible to traffic along the road.