You might like [Brian Conrad's handouts][1]. Specifically, [*Construction of tensor products*][2] and the two handouts after that one. They have a heavy emphasis on the universal property and examples.

(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