We encourage everyone near 60 to prepare for endgame by getting attuned to all raids as soon as possible.
Attunement to the Core can be picked up at lvl 55 outside the area that leads to Blackrock Depths (BRD) in Blackrock Mountain. You will need a run in BRD that goes as far as past Chest of the Seven and then kill 2 packs of elementals to get to the MC entrance and get your core.
To enter Onyxias Lair you need to hold Drakefire Amulet in your inventory. The questchain for this starts with a conversation with Warlord Goretooth, found in the tower in Kargath, Badlands. You can start this at lvl 55 and it opens a long questchain that will take you to Lower Blackrock Spire (LBRS) and through several runs in Upper Blackrock Spire (UBRS). For details see https://classic.wowhead.com/guides/onyxia-lair-attunement-classic
Your group will need to kill Scarshield Quartermaster, found in the entrance area of Blackrock Spire. He drops a quest item that requires you to kill the last boss in UBRS and then touch the orb behind him.