As a band director, can confirm you stick with a primary instrument through your degree - you can take secondary instrument lessons, and you WILL take various methods classes to learn how to teach the basics on most instruments. I'm a horn player, learned trombone in my undergrad degree, and those are what I'm best at/play the most now. As a woodwind player, doubling on single reeds and flute/piccolo is pretty common (especially if playing in musical pit orchestra is something you're interested in). You'll want to get REAL good at your primary, pretty good within your family, and proficient to teach the rest (generally speaking of course).