So, I'm watching ST:TNG

[vulcanlogician]

The Enterprise visits Deep Space Nine.

He's cute

My old roommate had a serious crush on Bishir.

As for "when does Wesley Crusher become annoying?" from the OP... I always thought Wesley was a decent character. In season 1 he is kind of one dimensional "Hey look! A kid on a starship! Whoa!" But I never thought he was annoying. 

I don't care for the way his story arc ends. The way that all went down kind of annoys me. 

But otherwise, I think he is a well developed character. Season 2 Wesley onward is not annoying in the slightest. The Traveler plot stuff is the only thing I could do without. And that's only two or three episodes if memory serves.