It's not instantaneous
If you tug one end, I don't actually feel the pull instantaneously. Your tug pulls on the fibres nearest to you, and the 'tug' propagates along the length of the string as the stretched fibres pull on the fibres behind them, in an attempt to even out the tension you've put upon the string.

Think about it this way: if you twang the string instead of tugging it, the small wave you create will propagate along the string, rather than reaching me instantaneously (of course, it will gradually get smaller, so the signal will be very small by the time it reaches me).

Doing this over a string of length 1 light minute, your message will take a very long time to reach me, if it gets there at all.
(LongJohnBaldry, Thu 27 May 2010, 9:56, Reply)