Dodger,
I would say that if you don't think strength is an issue for you in your hiking then work on some cardio. Start walking/running daily to build up your stamina, this is (I believe) why you get tired during long hikes. If you don't plan on doing any real rock/alpine climbing I don't think you need to worry so much about your strength training.
Use a stair climber or an elliptical a couple of times a week to work up your ability to take longer treks. If you don't have the ability to go to a gym or have gym equipment at home, jog up and down your stairs 10-15 times, or go out and take a run in your neighborhood. But like other hikers will tell you, nothing prepares you for a long trek like trekking itself.
Take some time to build up your hiking stamina by do intervals. For 2 weeks do about 5-8 miles, then the next two weeks do 10-15 miles, and so on until you reach your goal.
I hope this helps out.