New level 70

Depends on your class, but the core principle of course is to maximize stats gained per time spent.

Realistically, that means quests first: there's a lot of low-hanging fruit there, including some of pre-raid BiS items. Quests also come with an added benefit of also granting you reputation and gold, which you will need for flying (which in turn is required to access Tempest Keep dungeons, and subsequently, all raids besides Gruul's Lair). For example, I've made a list of pre-raid items for prot pally for personal use (https://pastebin.com/raw/YDJt2EiV) and you can actually become uncrittable with quest items alone (and while that is kinda unbalanced set, many of those items are very close to pre-raid BiS or actually are, of course depending on your other stats - Andormu's Tear for example is **** if you can get uncrushable without it). You should probably do world quests until you've exhausted the low-hanging fruit in quest rewards and purchased a flying mount (Netherstorm tends to be the best zone for gold).

The second order of business isn't as clear. You can start working on your professions (for example, crafted tailoring sets and BS weapons are extremely good), do specific normals for specific gear drops you need (for example, as restoration druid Idol of the Emerald Queen from Shadow Labyrinth first boss probably eclipses all other gear sources in importance), farm reputations for specific rep items you need, and once you have enough reputation to purchase the keys, do heroics. However, while you need Badges of Justice for some items, heroics are often time-inefficient given that you can only do each once a day, and if they are properly tuned, you can't clear most of them willy nilly. There's some heroics that have standout items (such as Quagmirran's Eye from Slave Pens and Sun Eater from Mechanar) while being among the easiest, those will most likely be the ones you'd want to farm for badges and do the tougher ones only once you've exhausted the low-hanging fruit elsewhere (or for fun, rep or quests).

And of course, Karazhan should be doable with just optimized questing gear and given the choice you should start doing it ASAP. The prequest starts in front of Karazhan ("Arcane Disturbances", you can get an optional breadcrumb quest "Archmage Alturus" from Lower City inn).