On an odd side-note concerning your wanted status in Markarth specifically and The Reach in general, if you start to fight the guards in Markarth and Vigilant Tyranus is still in town because you haven’t gone into the ghost house yet, he will join in attacking you with his mace and lightning spell- try to avoid this if possible, because you cannot successfully kill him, and he does markedly more damage than any of the guards do. Even more curiously, once beaten into submission, if the fight ends before he rises back to his feet he will immediately become nonhostile when he recovers, and may ask you again about the house.
At any rate, once you’re arrested, you’ll be stripped of all your possessions on hand, and when you arrive in the mine you will have some miner’s clothes and shoes and that’s it. If you had anything stolen in your inventory, that will vanish entirely and permanently, but anything else will be retrievable at the end of the quest, so don’t worry too much about anything you haven’t previously nicked off of someone else.
Your arrival in the mine is greeted by a large Orsimer woman by the name of Urzoga Gro-Shugurz. She runs the prison/mine, and explains to you that lying around isn’t the way things go here- you’re going to be mining as long as you last here, ‘until you start throwing up silver bars’. You can respond however you like- a flippant response gets a threat and asking when you get out amuses her, but regardless of your answer she has the gate opened to let you into the mine itself.
You’ll find yourself unable to do anything but move around until you’ve entered the mine and the gate is closed behind you- but once this happens, you can snag a little food from the sack by the front gate.
Down the ramp from here is the main chamber of Cidhna Mine, where a fire and a couple of other prisoners are waiting. These are two of the three most important prisoners here for the time of your stay, Uraccen and Borkul the Beast. Uraccen is the prisoner to go to for information on anything within the mine. When you first approach the man, he considers you warily and asks what you’re in for- you can respond by claiming your innocence (an idea he laughs at, both for how expectable it is and how true it likely isn’t), saying you’re in for murder (which gets a warning that other prisoners may consider this a challenge and try to fight you), or threateningly imply that you might want to kill him for poking his nose into your business (which prompts him to back off begrudgingly).