You might like [Brian Conrad's handouts for a sophomore differential geometry course][1]. Especially relevant are [*Construction of tensor products*][2] and the two handouts after that one. They have some nice examples and a heavy emphasis on the universal property.

(I don't think this warranted more than a comment, but I can't post those yet.)


  [1]: http://math.stanford.edu/~conrad/diffgeomPage/handouts.html
  [2]: http://math.stanford.edu/~conrad/diffgeomPage/handouts/tensorbasis.pdf