For a nominal fee, you buy a membership in a club. That is what we had to do this year to get tickets to a Chelsea match.
We did it right when the season started, but you could probably do it now. After you buy a membership (we paid about $30 each for the Chelsea memberships) you are able to buy tickets on what Chelsea calls their "ticket exchange." I believe that other top teams (Arsenal and Tottenham for sure) also do it the same way.
So, you buy your membership, then log into the exchange where season ticket holders who can't make it to a particular match put their seats up for sale. And you score your tickets there. No buying from touts on the street (and getting arrested, or kicked out of the stadium.)
Unlike in the States, the tickets on the Premier League team exchanges are NOT re-sold at a premium. We got ours for basically the same price that regular tickets would cost. And we didn't have to buy a "hospitality" package, which is CRAZY EXPENSIVE. We got decent seats, too.
You might think buying a membership in order to have the privilege of buying tickets is too expensive, but in fact, it is FAR less expensive than hospitality. You can generally buy a junior ticket along with an adult ticket, too, without a separate membership for each child. (Each club has their own rules, so double check.)
Also, we found that (at least at Chelsea's Stamford Bridge) the prices for a program (three pounds fifty) and a beer (five pounds) and a burger (seven pounds fifty) in the stadium were insanely less than buying food and beverages when seeing any kind of sporting match in America. So you will still come out ahead!