What's the best way to get gold without wasting time?

Given the Random nature of the Arena, I'd caution against trying to use it for Gold Gain -- at that point you are essentially gambling, and as such, it can not be relied on as a satisfactory general answer.

As to the others, Quests are the only real answer -- they're a guaranteed 40, 60, or even 100 gold, once completed, and you get a new one every day. While getting the 700 gold to open a wing would take 17.5 days if you only did 40g quests, it is possible to "re-spin" quests once a day by clicking the X on their corner, hoping for a 60g or 100g one instead.

The gold earned through Play Mode (10g / 3 wins) has a daily cap of 100g as well (30 wins) as an anti-cheating measure, so it cannot be abused to indefinitely gain gold.

So do quests every day, avoid the Arena unless you are okay with risking it all, and don't spend it on Card Packs -- you'll get that 700g eventually.

The safest way to make money is to do the quests and to remove the quests that aren't worth doing (40 golds) by rerolling them. List of quests.

Arena is too random to be a safe source of gold but can be a greater gain if you're able to win 6+ games each time. If you're confident you can do this, that would be the fastest way.

Doing either one will always get you some 10 golds for 3 wins once in a while but it doesn't compare to doing quests or arena as both provide better gold income.

The method I've come up with is to grind every 3 days. Spend a few minutes each day to log into Hearthstone and reroll any quests you don't want in between these play sessions. You should end up with 3 quests that can all be completed with a single class.

This method let's you work towards completing all three quests simultaneously, instead of winning 5 games as a Warlock for one quest and 5 games as a Priest for another quest. You won't get gold as quickly as if you went for the daily 100 gold earned after 30 wins, but you will get the most gold out of the time you spend playing. Plus you get more time to spend doing other things you love!

This method is also good if you want to level up a single class for Golden Class Cards or work towards the Golden Hero Portrait. I've been questing with my favorite Paladin deck for a while now and I am steadily working my way through Naxxramas. Just 350 more gold until Construct Quarter!