My apologies for the lengthy reply here…
Firstly, congratulations on reaching Level 70!
My usual advice in this kind of case is to do whatever you find enjoyable.
Like Zapp mentioned, it would be a good start to try and find a guild that is currently raiding/clearing Karazhan on a weekly basis (Most guilds do this for the badges, since it’s the highest badge-per-raid ratio in TBC at 22 per run, rather than acquiring the gear for their members - most loot is now disenchanted)
Healers and DPS have it much easier entering raiding content that they’re not quite prepared for than Tanks do. I’m not sure if you are DPS or Heals, but you could run some regular dungeons (Shattered Halls, Botanica, Steamvault, etc.) for the gear you need. Biggest thing is for DPS to aim for as close to your Hit Cap as possible (Casters = 16% / Physical = 9%)
I’d recommend looking at Seventyupgrades to track your current gear, and see what upgrades you can find through regular dungeons. It also shows you what gear you can receive through different faction reputations.
I would immediately express interest to the guild in raiding Karazhan and fill the time running dungeons until your first raid. Zul’Aman is another great option to start running as soon as possible because you can run it twice a week (compared to once a week for other raids) and I think it drops about 13 badges per run, so that could be 26 weekly!
At the end of the day, I wouldn’t put too much worry on getting any specific gear because the gear you will be able to obtain prior to Lich King’s release will likely be replaced fairly quickly. Do and get what you can, as it will make your life easier in the beginning, but don’t stress too much if you’re not seeing many drops.
As far as your Level 60 Druid…
For me, personally, I typically say that once you get your Dungeon Gear (“Pre-Raid BiS”) you will no longer need to run regular dungeons (unless you want to, of course) so you will be starting to focus on raids/heroics, and you can only do them so often. If your Druid interests you, I would recommend filling the downtime levelling it, absolutely.
Congratulations again, that’s awesome!