Old-world endgame requirements are as follows:
Blackrock Spire: No requirements for Lower; Upper requires a Seal of Ascension (one only). Seal chain starts with a drop in the Lower area.
Stratholme: No requirement for live side; one member must have a key for dead side (or else you can enter it after fighting a bit of live). Key drops from a boss on the dead side.
Scholomance: One member must have a key, obtained from a chain starting at Chillwind Camp.
Zul'Gurub: No requirements, but get the quests from Zandalar Isle (off NW coast of STV) for best results.
Molten Core: Entrance is at the end of BRD. Once you get there, a short attunement quest will allow you a shortcut.
Onyxia's Lair: One of only two that absolutely cannot be accessed without attunement. Loooooong attunement quest chain, begins from Morgan's Vigil.
Blackwing Lair: Entrance is somewhat past Rend in UBRS. A drop from just outside the BRS entrances begins an attunement quest to allow a shortcut.
Ahn'Qiraj: There's a world event to open it. Once that's happened (which it did long ago on Stonemaul, but has not quite been reached on Blade's Edge), no attunement is needed for either part.
Naxxramas: The other one that absolutely requires attunement (the "entrance" in Strat is fake). The attunement is purchased in Light's Hope Chapel, and requires at least Honored with Argent Dawn (free at Exalted, and much cheaper at Revered than at Honored).