With current ilevels from timeless, valor, etc, there's no challenge in getting 15% expertise, so ilevel is not relevant to the question of Caps>Haste vs. Haste>Caps. Short answer: Just get the caps and move on toward what really is going to matter...
Getting back into tanking, that is. You won't get any useful gear from dungeons, heroic or not, and since most everyone outgears then, I suggest sparing yourself the hassle of trying to hold threat on a 570 ilevel Mage blowing cooldowns. Skip them, and do relevant stuff.
1. It's not too late to get the legendary cloak, so get the
quest line started and it'll happen eventually. Good luck if you go for it. (btw, don't worry about the 500 stat gem. It only works in sha-weapons which are all very outdated.)
2. Hit the target dummy for 3-4 minutes at a time, at least once a day, just practicing the rotation. Generate and spend as much Holy Power as possible, and practice banking it, i.e. get to 5, use Shield of the Righteous (SotR) which gets you down to 2, get back up to 5, etc., that way if you see your health drop a bit during a fight, you can use SotR, generate 1 Holy Power, and use a second SotR right away, which helps mitigate those spikes.
3. Get
Striker's Battletags off of the Shado-Pan Assault Quartermaster in Townlong. It's only 825 Valor and doesn't need rep to get. Also consider getting
Lao-Chin's Liquid Courage off the vendor Shado-Pan Quartermaster. Even though it's a lower ilevel, it's a great starter trinket (better than the 496 ones for sure) and it only costs Justice points to buy. Having the 496 ones in your bags will get you a higher ilevel, but having Lao-Chin's equipped will get you a good chunk of health and a 1 minute mastery cooldown.
4. Heroic Scenarios sometimes drop 516 items (the first is guaranteed I think), but more importantly, it's a good source of valor and fun, especially with friends. They're fairly easy and forgiving, so a good place to learn how to put your abilities in use in various fights. They also have bonus Valor if you get the last parts done fast, so it's also a bit rewarding to get better at them.
5. Do Celestials every week for a chance at 553 Tier gear or 522 PvP gear. You don't need to do any stupendous amount of DPS or whatever, just hit that loot pinata. Kill rares on Timeless Isle and do the quests for easy valor and more chances at 535 upgrades.
6. When you feel comfortable staying alive (and I suggest getting 476+ weapon and shield first), get a healing friend and run ToT LFR. You'll be getting the quest items for
The Strength of One's Foes, earning rep with the Shado-Pan Assault which unlocks 522 ilevel gear (like
Vicious Talisman) available for Valor, and maybe getting some 502 gear that might be useful at first, but no biggie, there's better stuff out there.
7. If you can get through a ToT LFR without dying, then as soon as you have the ilevel to do SoO LFR, do it. Save your Warforged Seals for better gear from flex or normal though, except maybe one roll a week on Malkorok to get the
Vial of Living Corruption he drops.