Challenge modes can't be queued for like normal or heroic dungeons; you must travel to the instances and start them yourself (at least until you start getting golds and can teleport to them). Doing so will illuminate why so few people enjoyed having to do this in the age before the Dungeon Finder.
Right-click your character portrait, scroll down to instance difficulty, and click the Challenge Mode option. The instance portal will change to the hourglass graphic above. Enter and you'll find the path forward blocked by a yellow mist with a sundial in front and a list of the dungeon's requirements on your right. Clicking the sundial will give you the option to start the challenge. Once you accept, you'll get a countdown from 5, and then you're off to the races. Be sure to pre-pot!
For the first time you do every challenge, it's generally best not to reset for every mistake (which is something we did and regretted). Do the dungeon from beginning to end no matter how many wipes it takes. Learn the pulls, identify where you'll be using invisibility potions, and complete the objectives. A group that keeps pecking at a dungeon can manage all bronzes, and probably even silvers, without doing anything special.
How quickly can we finish all 9? Progress will depend on how prepared your group is and how quickly people get used to the CC and cooldown requirements that allow a group to move briskly. As always, the Schedule Boss is an omnipresent concern. Great is she, and terrible as the dawn.
My group worked on the weekends on and off for about a month. A plurality of our time was spent in the first two dungeons we did, which were Gate and Scholomance. Once we'd gotten gold in these two (and whenever we could get everyone online at the same time), the group knew what to expect and golds started raining from the skies. Experienced groups running challenge modes for valor or a paying customer can knock out all 9 golds in around three hours.
On the subject of bad nights: Sometimes due to a combination of circumstances beyond your control (someone doesn't show up, computers go boom, a key member keeps lagging, sunspots, each and every single one of you has by unspoken agreement decided to suck all at once), you will have an awful night. Just walk away and come back later. Banging your head against a wall won't do any good, and things usually turn out better when people have had some time to think about how to improve a run. Take a break, watch some videos, and come back for another try later.
Filed under: Analysis / Opinion