It depends on your class. But i prefer DungeonBuddy by far.
The best way that I have found is to use a healer for the short queue times to run dungeons, Heroics, and then LFR.
I've not ran any tanks in dungeons, it seems very risky. DPS work fine, the queue timers are just a bit longer.
tl;dr:
1. Assuming you quested 85-90... Buy Crafted PvP gear to get ilvl 435/460.
2. Run Heroics until geared decently.
3. Start LFR. Save up 1250 Valor for iLvl 22 Neck.
4. Buy some ilvl 476/496/522 BoE gear on the Auction House (depends on your budget and time line) to hit ilvl 480 to be able to LFR Throne of Thunder.
Cheers,
XtO