What do you do with old ציצית that do not fit anymore? Can you cut off the corners and throw the rest out respectfully?
According to the Shulchan Aruch (O"C 21) if the tzitzit are no longer valid, one may dispose of them. The Ram"a adds that one should dispose of them respectfully, but they do not require geniza. The Ram"a then brings down a second opinion that they (the strings) should be put in geniza, and recommends it. Similarly with the article of clothing itself: it may be disposed, but may not be used disrespectfully. The Mishnah Berura (S"K 2) adds that one may invalidate the tzitzit if he wants to dispose of them, but one should not dispose of tzitzit while they are kosher.
(Edit- According to the Mishnah Berura 15:3, one is permitted to remove tzitzit, even in a sense that renders them invalid, if the beged became worn out (or, presumably, it no longer fits and is not usable).)
In your situation, where the article of clothing no longer fits you, you may invalidate the strings and dispose of everything in a respectful manner. If you wish to follow the stringency of the Ram"a, you should put the strings in geniza.
See my answer here: https://judaism.stackexchange.com/a/12184/759
In short, tzitzit are not fundamentally holy and may be discarded in the trash, although it is proper to treat them with extra respect (ie wrapping it in a bag first or something). This applies to the strings. (Shulchan Aruch 21:1)
As for the beged (piece of clothing): in the standard case where the beged was designated specifically for the mitzva, then the beged can be thrown out normally but should not be used for other especially degrading uses . But if you had a beged that was not specifically for the mitzva but you wore it anyway as clothing, then no restictions would apply to it as it is not designated for the purpose of a mitzva. (Shulchan Aruch 21:2)