Ok, so here's what happened, in order:

 1. [Eru Ilúvatar][1] would create Elves and then Men in the coming of the [children of Iluvatar][2]. [Aulë][3], not wanting to wait until the coming of the children to have beings to whom he could teach his crafts, created 7 Dwarves in secret. But Aulë did not actually have the power to create life, so these Dwarves were not technically a sentient species.
 2. Ilúvatar finds out about these Dwarves and reprimands Aulë for his impatience. Aulë realises he has made a mistake and offers up the Dwarves for Ilúvatar to do with as he will, including destroy them if he so wished.
 3. Ilúvatar accepts Aulë's offering and grants the 7 Dwarves life, so when Aulë attempts to destroy his creations, they cower in fear and beg for mercy, which Aulë grants them.
 4. Ilúvatar decrees that he shall not suffer Dwarves to exist before his children, and so the 7 Dwarves shall sleep underground until his children awaken and walk Middle-Earth.
 5. [Yavanna][4] voices concern for her creations, the plants and animals, for she fears the Dwarves would fell them to fuel their industrious nature. Aulë, in defense of his Dwarves, informs her that Ilúvatar's children would also have need of trees. She then asks [Manwë][5] for counsel and asks that her creations be protected. When asked which she would have protection granted to, she replies that all things have their worth, but that trees should punish those who wrong them.
 6. Manwë informs Ilúvatar of Yavanna's request and Ilúvatar grants it. [It is said that when the Firstborn (The Elves) awaken][6], spirits shall descend to Middle-Earth and inhabit beings within it, guardians who will protect the plants and animals. This is the conception of both the Ents and the Great Eagles.

So in fact many races were created long before the Elves ever existed, but all were asleep before the Elves awoken. The first race to awaken on Middle-Earth were the Elves. Technically speaking, the very first race to "awaken" on Arda were the Dwarves, as they were created first by Aulë and granted life by Ilúvatar before they were put to sleep.

  [1]: http://lotr.wikia.com/wiki/Eru_Il%C3%BAvatar
  [2]: http://lotr.wikia.com/wiki/Children_of_Il%C3%BAvatar
  [3]: http://lotr.wikia.com/wiki/Aul%C3%AB
  [4]: http://lotr.wikia.com/wiki/Yavanna
  [5]: http://lotr.wikia.com/wiki/Manw%C3%AB
  [6]: http://lotr.wikia.com/wiki/Of_Aul%C3%AB_and_Yavanna