What is the quickest way to relieve sciatica pain?

Multiple ways. The sciatic nerve can be irritated because of a herniated disc, arthritis that results in tightening of the spinal canal, falls, injuries, or other rare causes (tumors, infections). Anti-inflammatory medications, steroids, physical therapy (especially with traction) can be helpful to relieve the pain. If these treatments aren't helpful, surgery may be indicated and may give the quickest relief.
Spine Pain Options. This chronic pain in the distribution as you suggested is the result of an irritated nerve or facet joints or other injury typically in the lumbar spine (low back) which are caused by herniated disks, spinal stenosis or degenerative disc disease, etc requiring further evaluation by a spine specialist and may be candidate for facet injections/radiofrequency ablation and epidural steroid injection.
No easy answer. Sciatica is a term that means pain radiating down the leg, usually to the foot, as a result of nerve compression in the spine. It is often self limiting, meaning it can go away on its own. As such, initial treatment is physical therapy and sometimes steroid injections, which can shorten the duration of symptoms. If that does not work, spine surgery can help alleviate leg symptoms.