Running dungeons without a Group Finder??
Running dungeons and raids will be quite different in World of Warcraft Classic, not only in terms of content, but in attempting to find a group to join you. The Dungeon Finder or “Looking for Dungeon” (LFD) system was not implemented until patch 3.3.0, and it will not be in Classic either.
This means that if you do not have any friends online and you are looking to run, say, The Deadmines, you have two options: First, sit in Westfall and spam the channels there with a message like, “LFM Deadmines, have a tank, need healer and DPS.” Or second, head to a busy city like Stormwind and spam the same message in the LFG channel there. You may find someone quickly, or you may spend an hour searching with no success.
Once you find group members, you then need to run all the way back to the dungeon entrance. There is no instant teleport like today. Instead, there will only be Meeting Stones, which still require two people to already have run there in the first place to summon the rest. If someone decides after beginning a dungeon that they need to leave, you will need to repeat the process. This was why completing a dungeon used to be considered a bigger deal if you used a Pick Up Group (PUG) for the whole thing.
Dungeons for the Horde, Dungeons for the Alliance
Geographical location is suddenly relevant again as certain dungeons were designated as "Horde" or "Alliance," at least in where they were placed. "Wailing Caverns" for example is close in level range to the "The Deadmines," but is easily accessible to the Horde, with the other being easily accessible to the Alliance. Players wishing to visit these dungeons are not restricted from doing so, but they will be in for a long run through enemy territory.
With this in mind, adding good, polite players to your friends list goes a long way. Playing as a healer or a tank does as well. This carries over today in the short queue times when using the LFD or LFR tool, compared to selecting only a DPS role. If you are a competent healer or tank, you will find yourself on a lot of players' lists. This is what players used to refer to in that WoW was a more social game during Vanilla, and they were right. The implementation of the group finder made it so you never had to talk to anyone if you did not want to, and if the group was not to your liking, leaving and re-queuing after your debuff wore off is all you need to do.
Need Before Greed
Blizzard has been clear in stating that loot trading of soulbound gear will only be available in endgame trading to help guilds ensure that the proper player receives an item and to cut down on requests for a GM to come and fix a problem.
Everyone else is going to have to deal with whatever happens in a dungeon. This means that if there is a beautifully-stated caster staff, one that would be a great upgrade to you, the only caster in the group, but you lost the Need roll to a hunter who accidentally (or “accidentally”) clicked on it, there is nothing that can be done. Do your best to not let it bother you, and try to group with people you trust.
The High Cost of Being Multi-talented
Finally, one last point about dungeons is the expensive, but unavoidable costs of resetting talents. As there is no dual-talent specialization in Classic, and the cost goes up to a certain maximum every time you reset your talents, Healers and Tanks may ask you or the party to cover that cost if they are coming as a favor to you.
Any players who remember Vanilla WoW can attest to the absurdity of attempting to level up as a Protection Warrior or Restoration Druid. They will likely be in a DPS talent spec, and sharing the costs among a party was, and will be again, another part of the game.
For a complete list of all the dungeons in Classic, check out Wowhead Classic. It will likely be a site players visit often as they slowly level their way through the multitude of quests that are available.