On the actual topic of conversation, it's only wierd if you make it weird. You CAN talk to girls in public places, e.g. transport, but you're going to look like a freak if you don't have a reason to bail.
E.g. Hey. Look, I'm going to go meet my friends in a sec, but I thought I might come and say hi ... blah blah ... we catch the same train ... blah blah ... Anyway, I might speak to you later.
It's not just you reciting a monologue, you've got to let her talk too.
Then bail like mad after a minute or 2.
Whatever you do, do not sit next to her. That will freak the shit out of her.
At this point in time she now knows you exist, but is not freaked by your prescence, as she knows you have friends and won't sit next to her freaking her out etc.
And wait a sec, next time you see her on the train you already have your opener. "Hey, we spoke last time etc."