when it comes to cars, e-bay is a mix of 'classified ads' where the expectation is the car may be advertised elsewhere as well, and any deal is outside e-bay, the seller just pays a flat fee whether it's sold or not; and sales through e-bay where they take a cut. going outside e-bay for the latter might save a few quid, but can be open to problems as you've got no come-back at all if it turns out to be a scammer.
as ever, buying from a dealer is going to cost a bit more than buying from a private seller, but comes with more chance of protection if something does go wrong and less likely to be an outright scam.
if you're after something very specific, there is likely to be a web forum for make / model owners and that will sometimes have cars being listed for sale.
i refuse to go near farcebook so can't comment on that.
depends how far you want to travel to get something. the last two times i've bought a car (i'm only on my 4th car after 30+ years of driving) i've not had a functioning car to get somewhere in, which has cut the choice of dealers (round here, quite a few are out in the arse end of nowhere because land is cheaper).
most dealers will be on auto trader, some will be on
exchange and mart, some will list on gumtree or e-bay. some will be on more than one. and dealers will have their own websites. i think i found the current puddymobile on auto trader rather than going direct to dealers' websites, but i did look at a few dealers that i knew were local and had been around for a while.
and cars are bloody expensive at the moment. again depends a bit where you are - on the fringes of london, the ULEZ thing has skewed the market, so petrol cars that meet the ULEZ standards have got more expensive, petrol cars that don't (and most diesels which don't) have got cheaper. i did seriously consider whether it was worth going back to diesel and accepting the ULEZ charge when i go to visit mum-tat...