Here is a very understandable introductory article by R. Sujatha. For a beginning student this is good.
In my case, after that article, my next encounter with motives was with the more precise definition of a motive from the initial parts of Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points”. It even sort of defines a mixed motive; in fact it is the only definition of mixed motive that I know.
Read the Mathscinet review and also, Jordan Ellenberg's opinion on this remarkable paper of Deligne. I myself was astonished when I first looked into it and saw how much stuff was contained in it.
Deligne's paper "Formes modulaires et représentations $l$-adiques" l$-adiques" proving that the Weil conjectures imply the Ramanujan conjecture, is almost close to the theory of motives even though it does not explicitly mention motives. Here the representations of the absolute Galois group on the étale, or rather on the $\ell$-adic, cohomology is considered. This might give some starting insight into the Galois representations approach to motives.
