If you truly have sciatica, you need to have the cause diagnosed. This is of vital importance to determining the best course of treatment. The best way to diagnose your problem is via neuroimaging (radiology) by either an MRI or CT myelogram.
An MRI is less invasive and completely painless. A CT myelogram involves putting a small spinal needle into the fluid sac around your spinal cord and injecting iodinated contrast into the fluid, then taking a CT scan.
You could see a neurologist, but he/she won't be able to do much for you. An orthopedist might offer you surgery, but in my experience with patients who've undergone back surgery--DON'T if you have any other choice. He/she may also offer you physical therapy, high doses of non-steroidal anti-inflammatory drugs, and various exercises. A neurosurgeon might also offer you surgery.
Depending on the cause of your pain, you may be able to get a minimally invasive therapy from an anesthesiologist or radiologist via an injection of steroids or local anesthetics at the site of the cause of your pain. Often it is related to the intervertebral discs in your back.
Again, the treatment depends on the cause. A neurologist is a good place to start if they order an MRI of your lumbar spine for you.
In short, that's what you really need first. Find a doctor who'll order that for you, and that's the first step. Even a family practice doctor can prescribe an MRI. Just make sure it's interpreted by a board certified radiologist--preferably one who is fellowship trained and has a CAQ in neuroradiology. Don't have a chiropractor, neurologist, or surgeon play radiologist by interpreting your MRI. It's not an easy thing to do.