What exercises can I do to mitigate the pain of sciatica?

Pain specialist 1st. Each person needs to be evaluated on an individual basis. There can be several reasons for "sciatica" or pain down the leg. Lumbar disc inflammation with or without nerve impingment, facet joint irritation, sacroiliac joint irritation, piriformis syndrome. A thorough physical exam and diagnostic studies will help determine what exercise will be best depending on your diagnosis.
See pain specialist. Look into a beginners hatha yoga class. It consists of a series of "poses" that works on strength and flexibility of the whole body but especially the "core". Doing stretches from a video or handout is like learning to be a doctor only from a book. You need someone to instruct you to make sure you are doing the exercises/stretches correctly. Otherwise, you may cause injury and defeat the purpose.
Core Exercises. Typically core strengthening exercises are the best to help relieve pressure and forces in the spine. I would try and google some or look them up on youtube to see descriptions. If you can do them yourself, wonderful otherwise ask your doctor to get a referral to a good physical therapist for further assistance.