There are 120 levels with increasingly tougher enemies.
The mines open up on the 5th day of spring in a new game when you recieve mail telling you that a landslide has been cleared up. Upon entering the mine for the first time you will meet Marlon who gives you a Rusty Sword.
The ladder down the hole will take you to level 1. Every 5 levels you will unlock a fast travel point by way of the elevator. Every 10 levels you will get a treasure chest (except level 30).
The path to the right (at the top) is blocked off by a big rock. This can be destroyed with a steel pickaxe or a cherry bomb. It will lead to the dwarf but you cant understand him until you find all 4 Dwarvish scrolls and donate them to Gunther at the museum & library.
The minecart transport system is unlocked by completing the boiler room bundles in the Community Center or purchased from the Joja Community Development Form. It will let you travel between the mine, blacksmith and bus stop. If you complete the crafts room bundles in the Community Center (or buy from Joja Community Development Form) you will unlock the Quarry which also has a mine cart transport station.
If you start to feel that your sword is not doing enough damage you can go see Marlon at the Adventurer's Guild to the right of the mines and buy a better one if you can afford it. You might also be lucky enough to find better gear in the mines.
At level 90 you will get the Obsidian Edge sword (30-45 dmg, -1 speed, +10 crit power).
Use the pickaxe to break any stones you want but you might also want to try using the hoe on any dirt patches you come across. Barrels and chests are best destroyed with your sword as using the sword does not consume stamina.
For a full list of all rewards from treasure chests and what enemies can be found on which level, check the wiki: http://stardewvalleywiki.com/The_Mines