Click Play, Custom Game, Join.
Click the Title Bar once to sort by name. All the ARAM ones will usually be at the top. Use refresh every 3 seconds or so because they update really often and apparently its too hard to dynamically update the # of players for the page you are looking at.
A proper ARAM will have Proving Grounds as the map, with random champ select (not blind). There are variations where you pick your champ blindly or draft (ABAM, ADAM).
When you join, check out the TIMER at the top. IF its less than 42 minutes, that means the host is a Tard and AFK or something. Drop that one immediately. (timer resets when someone Quits and back to the setup phase.)
Also if you see one where 1 side is filled up, then check the Invites off to the right side. If names match, that means someone has made a PREMADE team and is looking for people to stomp. You might wanna shy away from that as the random scrubs you get on your team wont be organized nearly as well. If someone on your team isn't level 30 then you are a slight disadvantage since they wont have the runes/masteries.
Dont bother setting your masteries,runes till 30 sec left. Someone will probably drop... theres an average of 5 drops per 1 game that gets started.