How to get into amateur motor sport?

My little contribution:

autotesting
tarmac carpark or grass field - around the cones against the clock; good fun and very competitive; large following; can use a standard car if you want but some are much more competitive than others; cost / effort: low
gymkhanas and autosolos are more or less the same


trials
ascending grassy hills, not against the clock so points are awarded for how high you can go; fairly large following; essentially three different categories and you can use standard cars in two of them; cost / effort: low

autocross
slimmed down version of rallycross (against the clock short races on grass or stubble fields); small following currently but undergoing a resurgence; good fun; can use an almost standard car if you want but the ground is a bit rough and there's some risk of damage; you need proper helment and overalls and a license; cost / effort: fairly low
there are a couple of events run on sand to the same format

sprints and hillclimbs (treated together as they're more or less the same)
against the clock on a short tarmac course; huge following although much of it is at the top end of the sport; competitive and challenging and quite high profile, but you don't get much driving for your money; can use a standard car; you need proper helment and overalls and a license; cost is quite high due to entry fees; effort low

drag racing
against the clock acceleration at specific drag venues; large following although much of it is at the top end of the sport; entry level category (run what ya brung) also popular with modifiers; good fun if you like straight lines; costs as much as you want really but entry fees can be low (£50)

trackdays
non-competitive driving days on racetrack or airfield circuits; large following as you get loads of driving; costs as much as you want really although you'll use a lot of fuel and tyres - entry fees are dependant on the venue you choose (typically £100-£250)

road rallying
road based competition judged against the clock and on navigational accuracy; you need a navigator; events occur at night; car needs to be fairly standard and there are strict rules; can be quite hard - puts some people off; fairly low cost and effort
Enduro rallies are the same; many historic rallies are to the road rally format (rather than stage rally format), although some are actually non-competitive 'tours'.
12-car rallies and scatters are entry levels of road rallying.

drifting
competition based on driving style / ability; large following although much of it is at the top of the field; cost and effort high as you need the right kind of car and won't get much out of it unless you spend a lot

rallysprints
effectively a sprint on a rally stage; only a few events run anually; you need a stage rally car; cost / effort high

autograss
similar to autocross but more race-biased; you need a special car but anything goes; cost low / effort medium to high; tends to attract a slightly traveller crowd (sorry if this causes offence!!!)

oval racing (Ministocks, rods etc)
similar format to autograss but in specific oval race venues; a bit more traveller

banger racing
speaks for itself - very traveller and not a good route into motorsport if you want any respect elsewhere!!

more expensive stuff
racing
stage rallying - can be cheap if you navigate
Minicross / rallycross - you might be able to do a short season for a couple of grand
karting (I don't know much about this so won't say anything)

Typically people follow one of four routes into car motorsport in the UK:
autocross / stocks&rods / banger racing - normally from a young age but it's frowned on by "the establishment"
sprints & hillclimbs
karting - enevitably leading to racing
rallying (often road then stage rallies, and sometimes via autotesting)

Each scene has it's own setup of clubs etc, but most local motor clubs like the ones you've found are into rallying or sometimes sprints / hillclimbs. Racing tends to be organised by circuit based clubs (eg BRSCC) or car clubs (eg Jaguar Enthusiasts Club).
Motor clubs vary, but the more active ones with large membership are generally good and will happily accept and encourage new members.
My own experience has been through 12-cars via a motor club, into road rallying and a bit of stage rallying / autotesting / autocross / trials and now trackdays. I've not spent much, have learnt a lot along the way (and made some good friends), and enjoyed it all. I'd advise a similar route, or perhaps going directly into sprinting and then circuit racing as funds allow.