Tenten's branch opens after completing Ino's quests and the first date with Sakura.
After the date, you'll see an "emergency case" option when talking to Sakura in the hospital. That will activate the Tenten's branch.
Talk to Sakura and go to the forest to look for a flower called "Fuchsia". You fail to find it so go home to sleep.
Next day go to the Ino's store. She says that Tenten may have the flower. Go to the Tenten's store. However, it's closed and there's a note on the door saying that she's at the training grounds.
Go to the forest and there you'll find Tenten at the meadow if you turn left. Talk to her and go home to sleep.
Next day go back to the forest and collect kunais and shurikens. Bring 5 kunais and 8 shurikens over to Tenten's to advance the story.
Enjoy the wondefrul animation with Tenten at the training grounds and after that you'll get the flower, Fuchsia. The flower must be given to Sakura.
Next day go to the forest and you'll meet there Karin that holds two vases. You tell her that this vases was stolen from Tenten's shop and offer her to re-buy it. Karin agrees, give vases to you and says that she will be waiting for the money in the Ichiraku. After that go to the meadow to find the Tenten and give vases to her.
At the evening go to the Ichiraku, where Teuchi will give you the note from Karin. Go to the training ground and rest for a few hours until Ino comes. If you promised Ino an extra training before, there will a sex scene.
You'll notice that someone have spying on you so you run after the spy. The chase will finish in the forest by the lake. There will be 3 ninjas that you have to fight. If you do something wrong, you will be beaten and Tenten's branch will be failed.
Correct choices: Bluff - Start a conversation - Make shadow clones - Keep talking - Try to evade - Try to use one of wrestling techniques
To be continued...