If you are playing a thief/assasssin type character, it is recommended you join the Thieves Guild as soon as possible.

The Thieves Guild is located in Riften. To get there, hire the carriage outside of Whiterun and ask him to take you to Riften. It will cost around 20 gold.

When you approach the gates of Riften, the guards will ask you to pay the Visitors tax. Ask them what the tax is for. They will tell you it's for the privilege of entering the city. Try to PERSUADE them that it is obviously a shakedown. If it works, they will unlock the gate and welcome you to Riften.

You run into a thug named Maul near the entrance. If you pay him some gold, he will give you information about the thieves guild. His brother Dirge works in their hideout. If you want to get in on that action, he tells you to find Brynjolf in the marketplace, who could probably use someone like you.

Continue going down the road, and arrive at the marketplace. Brynjolf will approach you and ask if you are running a little light on the pocket. Ask him how he knows. Tell him your wealth is none of his business. Then ask him what he has in mind. He will tell you he has an errand to perform, and needs an extra pair of hands. Ask him what you have to do. He will say that he is going to cause a distraction, and you are to steal Madesi's Silver Ring from a strongbox under his stand.

Once you have the ring, he wants you to place it in Brand-Shei's pocket without him noticing it. Ask him why he wants to plant the ring on Brand-Shei, and he will tell you there is someone who wants to see him out of business permanently. They've been contracted to make sure Brand-Shei remembers not to meddle in affairs that aren't his own. Since they are not the Dark Brotherhood, they are not going to kill him, just make sure he sits in the prisons for a few days.

You are going to have to sneak over to Madesi's stall and use your lockpick on the strongbox. Then, when you have the ring, you pickpocket  Brand-Shei and leave it behind.

Tell him  you are ready. This starts the quest, "A Chance Arrangement". He will tell you to wait until he makes the distraction, then show him what you are made of. Save the game before starting this, in case you get arrested and need to reload.

He will start his distraction, which involves having everyone gather around while he shows them some falmerblood elixir. While he is doing that, you need to act. Madesi had left his stand and came to listen to Brynjolf. So has Brand-Shei, a Dark Elf vendor.

First, steal Madesi's Ring. Go (in SNEAK mode) to his stand at the Southwest of the market. Make sure any guards that are patrolling have already passed. Pick the lock as soon as it says HIDDEN in your sneak status. You have to steal the ring quickly before a guard can notice you.  Lockpick the sliding door beneath from behind the stand. This is a novice lock, so should be easy to pick. To pick the lock, try first rotating the lock. If it does not rotate, then the pick is not in the correct place. Rotate the pick to another position and try rotating the lock again. If it rotates a little, then you are getting close. Make sure to stop rotating the lock if it seems stuck, or the pick will break. Rotate the pick again to a new position that will make the lock rotate more. When you find the right position and manage to rotate the lock 90 degrees,  you will have picked the lock successfully.

If you get caught, you could end   up in jail, pay the bounty, or fight the guards (not recommended). Most jails allow you to serve your time by sleeping in your cell's bed. You can also try to escape. Your equipment can be found in a personal belongings chest somewhere in the jail. You will still be able to complete the quest if this happens, by talking to Brynjolf again. But you will not get a reward and you will have the stolen goods confiscated. So if you do get caught, you should just load your last save.

Once you picklock the sliding door, pick Madesi's strongbox inside. It is recommended you are in first person mode to be able to best target the strongbox. The strongbox is also a novice level lock. It will contain the silver ring, and some more treasures. Feel free to help yourself to it all.

Sneak behind Brandshei. Pickpocket him. Go to your Apparel in the inventory, and choose Madesi's Ring, and GIVE it to him. Then speak to Brynjolf.

He will congratulate you for a job well done and tell you the Thieves Guild has had a run of bad luck and could use someone with your skills to help them.  Tell him you are up to it.

This completes the quest "A Chance Arrangement" and starts the quest "Taking Care of Business". Before starting on this new Thieves Guild Quest, feel free to hang around the market and watch Brand Shei being arrested for the theft of Madesi's Ring.