At First Sight: In the anime, Yahiko is practicing his sword techniques alone when his shoe (slipper?) rips. Tsubame appears and helps him mend it. And Yahiko kinda falls in love with her, although he refuses it when Kaoru and Sano tease him. But too bad the manga doesn't show how they first meet. It only shows that when Yahiko goes to work at the Akabeko, he meets Tsubame who also works there. What happens next is in the Relationship paragraph. 

Relationship: We don't get to see much of them. Well, this is what happens: A bunch of gangsters threatens Tsubame to give them the key of the Akabeko, so they could easily rob it. Yahiko tries to stop them but they beat him up pretty badly at first. After getting a bit of advice from Kenshin, he's able to stop the thieves finally. The gangsters never bother Tsubame again... Puppy love? Their relationship may seem nothing serious, but it's gonna develop into something in the future, I'm pretty sure of that.