Climbing stairs is probably one of the best exercises you can do when preparing for hiking. You can do them outdoors or indoors (in a tall building if available). Even better, add a backpack with some extra weight when doing stairs and if possible wear the same hiking shoes as if you were hiking. You can also do some uphill runs, uphill jumping squats and lunges going uphill as well. Another option would be to go for some runs on the sand or grass to increase the strength and stamina in your legs. Again, if you can perform these exercises with a backpack it will be great.
When walking around and running errands, try to carry a backpack with you so you can create similar conditions to your hiking. There are many more ways to train for a hiking trip, but in my opinion I believe these are few good options for you to follow.
I hope this helps!
I also add in balance work or training on an unstable surface. Hiking usually involves walking on changing or different surfaces (rocks, uphill, downhill, over a stream, mud, etc). The chances of slipping, sliding, or losing ones balance is magnified due to the changing surfaces. Balance work helps to prepare those smaller muscles and the core for those changes.
I agree with stair stepping. Step ups for time are great as well. I will do them for up to 10 minutes (that is the limit before boredom sets in). You can progress to using a weighted backpack as well. Incline treadmill walking seems to emphasize the lower legs moreso than step ups and is great for preparing for those long steady climbs.