I am brand new to D&D (5e), and am starting to build my character.
Is there a way to be a sprite-genie hybrid PC? If so, how would that work, and could I grant wishes?
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A Sprite and Genie both exist as creatures in the D&D 5e Monster Manual, but are not viable PC races. A Hybrid of these two things does not exist in the rules.
As well, granting wishes is an extremely high-level spell (it's even described as “the mightiest spell a mortal creature can cast”), and it is unlikely your character would be able to do so early in the game.
I would recommend you check out the Genasi player race (from the free-to-download Elemental Evil Player's Companion) as this is most similar to what you have described. There is also a warlock subclass in Unearthed Arcana (playtest material) based around being subservient to a Genie, which may fit your character as well. Something to discuss with your DM would be reskinning this class to fit what you are describing, while still remaining in the bounds of the game rules.
Almost anything is possible in D&D, and if you search hard enough, you just might find a homebrew race that fits your concept... or you can invent something yourself.
However, for a first-time player (& DM), I strongly recommend that you stick to official published content.
The closest official published race is the Genasi, as Jon suggested in his answer. They are half-genies. There are four subtypes, one for each element. Since you want something spritely, perhaps an Air Genasi would be your best choice - they get a bonus to dexterity, and can levitate once a day.