I am in my early 20s and have asymmetrical bags under my eyes. What is the best way to go about fixing them? (photos)

The best long-term treatment for under eye bags is transconjunctival lower blepharoplasty with fat/bags redistribution (where the fat is partially removed and partially moved to fill the relative hollow area below it, in order to avoid sunken look). See following video (graphic) and website link below. Another option is filler injection.

You may want to consider a combination of treatments to get the best results.  Fillers can be used to add volume to hollow areas, revealing a smoother look.  Peels, lasers, and microneedling (Skin Pen) can be used to improve the quality and texture of the skin.  You may want to try these options before surgery, depending how aggressive you want to be.

There are generally three methods to reduce lower eyelid bags and wrinkles.
1. Utilizing peels, lasers or any other ablative or semi ablative techniques to stimulate new collagen growth. Though effective, most people want to have little to no downtime and in order to get a very effective result with which most patients would be extremely satisfied, these techniques will fall short of their stated goals. These techniques are wonderful if they are utilized in an appropriate strength. 
2. Carefully injecting a temporary filler (such as Belotero® or Restylane®) underneath the skin in hollow areas with magnification. This could be quite effective but does involve the risk of producing irregularities and lumps and will necessitate a certain amount of bruising. A permanent filler such as Bellafill ® or even a fat transfer utilizing purified fat from your body are also options.
 3. Performing a simple surgery under local anesthesia called a pinch blepharoplasty. In this operation, a small strip of skin is removed directly underneath the lower eyelash line and results in a nearly invisible scar. The recovery is very rapid (a couple of days) and involves one single stitch which is removed in five days. This approach is actually the least complicated in appropriate hands and will produce the most consistent and significant of results in reducing your lower eyelid bags and wrinkles.