A father has an obligation to teach his sons Torah, a parnassah and to swim. To what extent do these obligations extend to his daughters too?
(edited to include my explanation from the comments)
As Ariel mentions in his answer, Gemara Kiddushin 29-30 discusses the 6 obligations a father has to his sons.
1 Bris, 2 Redeem the firstborn, 3 Teach him Torah, 4 find a wife for him, 5 teach him a parnassah, 6 (some say) to teach him to swim
1 and 2 necessarily apply only to sons not daughters. We learn from a pasuk that 3 applies only to sons not daughters. 4 applies to both sons and daughters (although in different ways). 6 is learned from the logic that his life may depend upon it, given that I don't see any reasonable way you could say that this shouldn't apply just as much to daughters as to sons. You can't really say that maybe at some future time your son might drown if he doesn't know how to swim, but if your daughter drowns that's OK.
The real question seems to me to be by Parnassah. The Gemara doesn't make clear whether or not this applies to daughters as well as sons.
The Gemara brings two reasons. In the first Chizkiyah shows a Pasuk that compares livelihood to getting married. So just as a Father has an obligation to see his son married, so to does he have an obligation to teach him a Parnassah. This would imply that this would apply to daughters too, since the Pasuk on marrying off your children explicitly includes daughters.
The second reason the Gemara gives for the requirement to teach your son a Parnassah is to compare it to the obligation to teach Torah. This would imply that the obligation would only extend to sons and not to daughters since the pasuk for teaching Torah excludes daughters.
Based only on my own reasoning, I would think that this obligation would mean that you should teach your children what they need to know to survive in the society that they are in. If they live in a society where women normally cook and sew and such then you should make sure that they learn this. If they live in a society where the wife will be expected to earn a living with a job outside the home then I would think that Rabbi Yehudah's rule that not teaching a trade is teaching to be a thief would apply. I want to make clear that I have no source for this.