Genies perfectly exist! On spiritual plane. They are non-physical entities.
Genies do not grant only 3 wishes, they grant as many as you want. In wishes though, there's always a price, ususally it's energy... So essentially everytime you ask for a wish, they are going to ask something in return.
Genies are not like TV generally, you cannot bind them, neither be their master and so on.
Jinni is the singular term for the word Jinn. They originate or at least feature in Islamic literature. In the Quran they are beings with free will like humans and unlike the Angels meaning they can be "good" or "bad."
I'd suggest researching the Jinn to gain a better understanding of them and how one would interact with one.
Same way you would find any other spirit. Research it, find out how it can be summoned and if you are able to do so safely. If you can then summon it, if not keep researching, see if there is another way to find one.
The Jinn do not ''grant wishes'' any more than other spirits do.
And the three wishes thing is also fairly new of a concept.
So who/what are the jinn? Well, put simply the Jinn are a variety of tribes of human-like spirits. Their magic is usually quick and dirty.
As for how to contact them, you'll find a variety of instructions in a variety of different places. If you thought the ceremonialism of Western magicians was a bit ''much'', the Muslim magi actually made giant iron circles before summoning jinn, and made sure it was big enough for them in case they fainted.
However, to meet a Jinn, an easier road than the ''say its name 2412412541 times'' would be to simply ask an ally to introduce you to one, or go look for one on the astral plane yourself.
Also, just because something resides on another plane, does not mean it can't be summoned. While one can argue on whether they are summoned into the physical world, or the corner of the astral plane you duplicate and have trained your senses into perceiving, it is irrelevant, as they are there.
In fact, summoning something from the astral plane is easier than summoning something from the physical one. I mean imagine standing in a circle and whistling hoping your dog will come.