It would help to see a picture, but here are some general principles. If the under eye bags get worse when you gently press on the eyeball via the upper eyelid, then they are related to forward protrusion of the eyelid fat compartments. These can be approached either through a transconjuctival incision within the eyelid or externally through an incision beneath the lashes. Depending on the anatomy the fat can be either sculpted or repositioned
If the lower lid puffiness is more superficial, it will not respond to pressure via the upper eyelid. This swelling is usually related to allergies or other medical issues and doesn't respond to surgery.
Discoloration of the lower lid skin may be related to diminished skin tone, in which case it will look better when you gently push the lower eyelid skin in an upward and outward direction. If so, a tightening of the skin and possibly the muscle of the lower eyelid will help. If this is not the case it is probably related to the natural pigment of the lower eyelid and will not benefit very much from surgery.
I hope this information helps.