become members of their official supporters' group. That way members of rival fan bases can't get tickets with the home fans. Yes, the fans are totally and completely segregated at every EPL match. Then, once you've become a member of the supporters' group, you get the tickets that your seniority (really the lack of seniority) allows. It makes sense. The season ticket holders and fans who travel to away games have accumulated enough priority points so they get first dibs at any of the tickets.. this is in order for the atmosphere at home games to continue to be good (visiting fans sit on their hands).
I've bought tickets a few ways.. through the supporters' group, through third-party sellers like StubHub (which the British absolutely hate because it prices the average fan out of attending games) and from scalpers (they call them touts and also hate them for the same reason).
I've gone to a game and not had tickets thinking I could get a single ticket from a scalper.. wasn't able to. The game started and everything outside the stadium was a ghost town. Remarkable, actually.