Ret/Mage is certainly a viable comp. If it is the first time you're trying out arena, don't get discouraged after losing 90% of your matches.
Firstly, double DPS gameplay is much less forgiving and harder to execute than healer/DPS (unless you play double BM hunter ofc) as you have much less room for mistakes and need great coordination. Secondly, your gear probably sucks and you will therefore get stomped often no matter how bad your opponents are, because you just won't be able to score a kill in a certain window of opportunity and there will be nothing to do about it.
As per gameplay. Mage starts Invisible, Ret pops behind a pillar and you wait for a sec. If the enemy is charging the ret, you try to separate them so one gets caught behind a pillar and the other one preferably polymorphed in the open (i.e. out of line of sight of each other). On the other hand, if they pillar hug, you position yourself well (that's probably the most important part) for an opener and as ret charges in, the mage opens on one target with frostbomb deep freeze, trying to possibly catch both enemies in the pet nova. If there's a mage against you, don't pop invis sooner than he does, unless he's charging you. Then the ret should try to get him out so you can keep your invis.
The ret should spec into Fist of Justice (as Repentance shares DR with polymorph and a 50% slow is useless with all of the mage's control and snares) and always use the stun on the non-kill target (it DRs with Deep Freeze and you don't want that) in the opener. That either forces a trinket right away or opens up the target for polymorph, which will definitely force a trinket.
The ret should pretty much pop all his cooldowns right at the start to keep pressure, because once a double DPS loses pressure and has to go defensive, it's pretty much over. Use the opener to your advantage as much as possible.
Stealthers tend to fuck openers up so I'm not gonna go too much into detail on how to deal with rogues. It's extremely situational.
The mage should preferrably spec blazing speed against double melee and PoM against everything else for those clutch PoM polys or extra dmg in a deep freeze. Don't forget that the mage can Alter Time PoM so you can use it twice. Nothing like PoM polying a healer who just trinketed a PoM poly.
And a few tips:
DKs: blanket them as you deep freeze them, they won't be able to Icebound out of the stun.
Hunters: Try to CC the stampede with frost nova/pet nova/ice ward to negate most of the damage and pushback.
Rogues: Try to go on the other target while CCing the rogue to get his trinket and cloak. Without those, he dies in a deep freeze.
Warlocks: Don't let them put up gate at all costs!
Mages: Invis later than the enemy mage.
Ferals: They slaughter mages. Try to put more pressure on them so they have to play defensively. If you let a feral with CDs stick to a mage, it's over.
Paladins: After bubble, they're dead meat.
Oh and good luck and don't give up. It'll take some games to get the hang of it. Try to communicate as much as possible. Tell your partner everything you do and see. Try to come up with stuff in advance (i.e. instead of "Deepin X" say "In 5s I'll Deep X"). And, have fun.
I don't want to go back to fighting a healer's blue bar against half the comps.
Not necessarily a bad thing as long as the blue bar doesn't last more than 3-8 minutes depending on effective cooldown usage and mana efficiency of the player. Lowering passive regen substantially and also letting drinks give mana back sooner than after falling asleep at the keyboard would definitely promote a more active and entertaining playstyle for healers. And it would actually make sense to actually bind the "efficient" heals again. I don't even remember the last time I used heal/nourish/holy light etc.
But yeah, PvE ramifications... Still, if there is a -healing done aura, there could be a -regen aura in arena too imo.