I get a weird throbbing/stabbing feeling in my forehead every time I move my eyes. What sort of doctor should I seek out?
Ophthalmologist.
My wife had something similar and it turned out to be a type of neuralgia. Unfortunately, not much can be done, other than not touching the eye.
In your case, perhaps never moving the eye? 
Hope it is nothing serious.
If it were me, Emergency Room doctor. That sounds awful. If you think you can wait, opthamologist.
You aren’t providing enough information about your complaints to make an informed recommendation. Besides that would probably be construed as ‘seeking medical advice’.
In general, eye complaints are best evaluated by an ophthalmologist or an optometrist. That being said, non-ocular processes can present with ocular complaints.
How about seeing a primary care doctor, like your GP, before rushing off to see a specialist?
This isn’t the first time I’ve seen this question here, is it a common thing in the US to book your own appointments with specialists without a referral from a general practitioner?
I would have thought that that wastes a lot of doctor’s hours and delays diagnosis considerably.
That’s what I’ll probably do, but my experience has been that primary care doctors aren’t very good at figuring out what specialist to send you to, and tend to ignore symptoms that they haven’t seen very often. I’m soliciting advice so that I can be well-prepared when I go to the doctor.
If someone in the US has their health insurance through a Health Maintenance Organization (HMO), they are usually required to get a referral to see a specialist. Sometimes this is a royal pain, when the need for a particular specialist is obvious. For a while, some of them were requiring women to get referrals to see the OB/Gyn. That particular referral is generally not required anymore, though.
If someone in the US has another type of insurance, or no insurance, a referral generally isn’t required. It’s the patient’s choice. It might save money to go straight to the specialist, since only one appointment will need to be paid for. OTOH, if it’s something the GP could have handled, the specialist will probably charge more.
In the case of the OP, an urgent care center seems like a good choice. No need to wait for an appointment, can get some serious concerns ruled out, and may get the appropriate treatment or referral.
My current insurance allows me to see any specialist in their network without a referral. For some ailments, this definitely saves time and money. For example, I have had skin problems that required a dermatologist, and I didn’t need a GP to tell me that.
Are you having any other symptoms? Sensitivity to light, or some other thing?
I would not recommend an urgent care- the most a doctor can do for you in a facility like that is look into your eye. If he sees something, he’ll send you to a specialist or the ER. If he sees nothing obvious, he’ll send you to a specialist or the ER. What’s an urgent care doctor going to do for you?
I’d recommend an opthamologist or a neurologist. If you can’t see a specialist right away, see your GP first to rule out something emergent.