I play alto saxophone, piano, ukulele, and guitar. I am wondering if it's possible to take up piccolo as well, without playing the flute. I plan on taking lessons, as there are teachers in my area, but I don't know if I should start on flute first because I know piccolo is an extension of flute. I don't plan on playing it professionally, just for fun. Thanks a bunch!
Depends on your frustration tolerance. Piccolo has a more challenging embouchure than flute, but you know what? Fife has a much more demanding embouchure than either, and I learned fife first. This meant it was about six weeks of assiduous blowing making nothing but a "wwwwfffff" sound, before I actually got a noise out of it. Another week or two before I could generate that sound consistently. And then, trying to make it musical. Well, as musical as fife gets.
I have rather a lot of patience -- or at least did when I was 12. Six weeks is a long time to go attempting to do something and getting no positive feedback of progress whatsoever. And I can't guarantee you'll get it in six weeks (though piccolo is easier than fife, and I had no advantages at all, except maybe stubbornness, so I wouldn't be surprised if it takes rather less than that).
So, if you think your resolve to learn piccolo can survive a month+ of determined striving with no positive feedback to encourage you along, then go for it.
(For what it's worth, I was taking that class with a friend who played flute. The only one of us with traverso experience, she, the class champ, managed to get sounds out of her fife in about... three weeks.)
From the perspective of a music educator: the flute would be much easier to learn proper embouchure fundamentals on, as, with the lower pitch, it's more more forgiving. During my college days, I picked up the flute quite quickly, but I had a difficult time achieving a good tone on the piccolo. That said, it would not be impossible to learn on it's own, just more difficult.
You can find cheap flutes at pawnshops or on craigslist. You may try picking up a cheap one to begin on. If you get a used name-brand student flute and take care of it, it will retain value well. You could then sell it when you buy your piccolo, or just keep it around.