From what you are describing and from the photo it seems that you have two issues. 1) eye bag 2) superficial veins.
Scarless Eyebag surgery is one of the options to remove your eye bags and can be very effective if done by a proper plastic surgeon. The downside, of course, is the cost and downtime post-operatively. Another thing to keep in mind is that eye bag surgery may not treat those tiny superficial veins we see in the photo. And lastly, it is very important that the surgeon during the eye bag surgery remove similar and correct amount of fat on both eyebags.
An alternative, with next to no downtime for your consideration would include a combination of fillers, HIFU (High intensity focused ultrasound), threads for the eyebags and lasers for the superficial veins.
I don't think there is a "best way" but it really depends on what you want and for you to weigh the pros and cons of each style of management.
I hope this has helped a little bit in your decision making.
