There's not anything you can do on a DIY basis, it happens to most of them (and the closely related AE-1). But you might be able to get it working temporarily by sitting the camera on top of a dry towel on top of a radiator or after placing it in a warm airing cupboard for 24 hours; this may soften up gummed lubricants enough to allow the parts to move as intended. If it works, this just leaves the camera in a better state to be assessed for a proper repair, it's not a true fix and will go wrong again.
The camera usually just needs a good CLA (Clean, Lubricate, Adjust) service by a professional camera repairer. This should include checking and adjusting the light meter calibration at the same time. Afterwards it should ideally be serviced every year or two, but will probably last another thirty or so years!
In the UK expect to pay at least £60 and about another £15 to £20 if you need the one-off repair of having all the gooey foam light seals/mirror buffers replaced. You can try haggling the price, but as there is far more demand than supply I doubt that you'll find the job much cheaper. If it is, corners ARE being cut: I've seen some cameras supposedly "fresh" from a CLA which have simply been cleaned, polished and had a little bit of WD40 or 3-in1 oil sprayed inside. Almost invariably these were done by less reputable folks on auction websites for around £40.
I use a number of repairers worldwide, but if you're in the UK then I highly recommend
Colchester Camera Repair Service.
It's fair to advise that the repair will cost more than the camera is currently worth, but you can't buy a fully working and freshly serviced SLR for less than the repair cost, so it's definitely a worthwhile investment.
Good luck, you have an excellent camera with a great supply of excellent and relatively cheap lenses which can easily outmatch the photographic quality of current digital SLR's if used correctly.
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