Wand cores are crucial, as without a core from some magical creature, the wand would be but a stick. 

Unicorn hair generally produces the most consistent magic, and is least subject to fluctuations and blockages. Wands with unicorn cores are generally the most difficult to turn to the Dark Arts. They are the most faithful of all wands, and usually remain strongly attached to their first owner, irrespective of whether he or she was an accomplished witch or wizard.

As a rule, dragon heartstrings produce wands with the most power, and which are capable of the most flamboyant spells. Dragon wands tend to learn more quickly than other types. While they can change allegiance if won from their original master, they always bond strongly with the current owner.

Phoenix feathers are capable of the greatest range of magic, though they may take longer than either unicorn or dragon cores to reveal this. They show the most initiative, sometimes acting of their own accord, a quality that many witches and wizards dislike.

I would suggest using just these three, as they produce the best results.