Having to carry heavy luggage can indeed be very stressful, but there are some tips to help lighten the burden.

Look for a Sturdy Light Bag with Wheels and a Handle
It's bad enough that you have all that heavy luggage to convey, but you can make it easier to convey by doing so in a relatively light bag with wheels and a handle. Using a light bag for your heavy luggage can help take some extra weight off when your bag is weighed for check-in, and can help you keep your excess luggage weight within a manageable range.

Mind the Way You Lift the Luggage
If you have to lift heavy luggage, do so by bending at your knees not your waist. Then, bear the weight by lifting with your leg muscles rather than with your back, lift it close to your body while grasping the handle properly and then straighten up to put the luggage at the desired location. This way of lifting heavy luggage is unlikely to have the kind of adverse effects on your physiology that lifting it the wrong way will have. Additionally, when placing your luggage in an overhead compartment, first lift it onto the top of the seat, before firmly placing your hands on the left and right sides of the suitcase and then lifting up. If the luggage has wheels, ensure the wheels are set in the compartment first, before pushing the rest of the bag into the compartment.

Don't Twist Your Body When Lifting Heavy Luggage
Twisting your body when lifting heavy luggage can exert a devastating amount of stress on your upper and lower back that can result in subsequent back pains. It is much better to first point your toes in the direction you intend to turn to, before you turn your entire body in that direction. You should also not rush yourself when carrying heavy luggage, it is best to take your time and if it is too heavy for you to bear, ask for help or set the luggage down for sometime, to rest, before lifting it again.

If You're Using a Backpack, Never Sling it Over One Shoulder
Slinging a heavy backpack over one shoulder does not allow the weight of the bag to be distributed evenly, it thus leads to muscle strain and you stressing your back much more than you should.