I know that gannets aren't rare, but for a beginner birder like me, who hasn't seen a wide variety of birds like most of the people here have, that's what I consider rare, because those were the only ones I've seen as a beginner birder. And I just want to clarify that I was asking about what birds occur in the UK that are considered "rare" but that's my fault for not making that clearer.
Well, of the resident birds, apart from failing introduced pheasants and suchlike, naturally scarce (but perhaps near the carrying capacity of their current UK range) would be things like Golden Eagle. One of the rarest birds of prey would be the heavily illegally persecuted Hen Harrier, but that remains more numerous than the edge of range Montagu's Harrier. Some wetland birds like Bittern are nowhere near as common as they should be but are recovering in the recent absence of persecution.
Chough might fall within a definition like "birds that occur in the UK that are considered "rare"", but Gannets can't be considered rare under any reasonable definition, however few you have seen. They are a common UK seabird that is difficult to miss on a short seawatch on any part of the coast at almost any time of the year.
Reading some of the species related threads under Conservation on here may help you discover which are the scarcer birds of Britain. There are quite a lot that are in real trouble - modern intensive farming has caused local to regional extinctions of birds such as Corn Bunting, Tree Sparrow, Grey Partridge, and some woodland species are also quite likely on the way out due to habitat fragmentation: Willow Tit, seemingly also Marsh Tit, and Lesser Spotted Woodpecker have all crashed lately with recovery apparently unlikely. Summer visiting Turtle Doves are quite simply being shot out of existence on migration (as a birder it's often difficult to love our fellow humans!)
There is plenty of material on the internet generally, too: conservation organisations like the county trusts and RSPB will have details and even wikipedia entries for countries and species often contain details on distribution and population. Finally nature programmes from David Attenborough epics to Springwatch and its seasonal sisters abound with such information as part of their intent to bring conservation issues to the public.
So you are tapping into a hobby where the information you want is available in vast quantities. I hope getting rather more than you bargained for on here - and not quite in the form you perhaps expected, including from me - doesn't put you off what can be a lifetime of enjoyment.
Cheers
John