Intimidate Garl for her release (45% Speech required).
Quietly kill the two guards at the rear of the building and pick the lock on Tandi's cell.
With a Luck of 9 (and male gender, and wearing leather armor or using a Stealth Boy), the raiders will mistake the Vault Dweller for Garl's father whom he killed to take control of the Khans (50% chance). With at least 6 Intelligence and a Charisma or Speech check, the Vault Dweller can bluff Garl with this ruse and secure Tandi's release.
Use dynamite on the door to her cell.
The first and second methods of rescue entail a lot of risk, with the second method especially so should the Vault Dweller still be at a low level, as Garl is a highly competent brawler and is much more experienced than his men.
Tandi can be used as a party member if the Vault Dweller never takes her back to Shady Sands. However, since she cannot be given combat commands and doesn't appear to change her equipment she is of limited use. She also doesn't have comments for any area or event beyond just taking her home, whereas the other party members will comment on various areas, buildings and people throughout the game.
Asking Seth for money will make him behave aggressively, but this will not make him hostile. Though, should the Vault Dweller say they will not do the job without hearing about the reward, Seth will become angry and turn hostile (along with nearby guards and Katrina). While it is possible to escape to the world map, this effectively locks the Vault Dweller out of Shady Sands, as Seth remains hostile.
Killing the Khans before Tandi is kidnapped will rule out this quest. Likewise, if the quest Join the Khans is completed, the same will happen.
When escaping from the base with Tandi, a note will appear: "You gain 0 experience points for freeing Tandi." (Only for the violent solution. For freeing her non-violently, you will receive 400 XP.)
If the Vault Dweller has Ian as a companion, and fights Garl, he may get into the ring. If this happens, he will attack Garl; (Unarmed or by shooting as well) when Garl does not. This is of great help in completing the quest. Normally Ian is outside the ring and cannot help. [verified]