I’m leaving

The most idiomatic way to say “I’m leaving”, appropriate in most contexts, is probably:

Je m’en vais.

or simply

J’y vais.

It’s also common to say “I’m going to leave” rather than “I’m leaving” even if you are already in the process of leaving:

Je vais y aller.

(The last two somehow sound softer than “Je m’en vais”, which could even be slightly hostile, depending on the tone.)

“Partir”

Depending on the exact context,

Je pars

is also good, perhaps a bit more formal. It would feel most natural in a dialogue like this one:

Pourquoi tu mets ton manteau ?

Je pars !

or for a long journey so that “I’m leaving for Hong Kong tomorrow” would be

Je pars pour Hong Kong demain.

As mentioned by @GAMPUB, it’s also possible to signal your intent to leave very soon using “partir” in the past tense.

Je suis parti.

It’s a bit surprising on the face of it, since you obviously aren’t gone when you say that, but the implication is that you are already late/not available to talk or do something anymore. I would not recommend using it as a language learner, though, it might not always fit the situation.

Similarly, you could sometimes say “I’m not here”/“I’m already gone”:

Je ne suis déjà plus là.

Another odd use of “partir” in this context is

Je me sens partir

which literally means “I feel myself leaving” and can be used in colloquial speech to say that someone wants/intends to leave soon.

Parting with a group

It’s also quite common to leave a group by saying

Je vous quitte

Je vous laisse

Je vous dit au revoir

In that case, the emphasis is on the fact that you are leaving the group, not so much that you are leaving a place. In some contexts (say you have to do something or see someone else and won’t get a chance to catch up later), you might even say that and stay in the vicinity.

“Sortir”

Your third option,

Je sors

is a bit more difficult to use correctly. Often, there is an implication that you are now in your normal abode and will come back. It’s probably not appropriate at an outdoor location.

It’s perhaps closer to the English “I’m going out”, including when talking about a night out so that “I’m going out tonight, I have been invited to the restaurant” would be:

Je sors ce soir, on m’a invité au restaurant.

And, together with “avec”, it implies a sexual/romantic relationship but not a deep love/long-term relationship (a bit like “to date”, although not quite, as I think “sortir” means you have a relationship, not that you are merely flirting/hoping for one as “dating” sometimes imply in English):

Elle sort avec Nicolas.

“Sortir” is also what you would use when speaking about leaving a vehicle, but not necessarily with the intent of parting with your company:

Je sors de la voiture, tu viens ?

Slang

Beyond that, there are literally dozens of other ways to say “I’m leaving” in slang

Je me sauve

Je file

Je décolle

Je disparais

Je me casse

Je me barre

Je me tire

Je fous le camp

Je fiche le camp

Je dégage

Je plie bagages

Je m’arrache

Je bouge

Je prends la tangente

Je trace [la route]

Je [me] taille

Je cargue la voile

Je m’envole

Je décarre

Je décampe

Je vide les lieux

Je gicle

Je trisse

Je jarte

Je me translate

Je m’exporte

Je déguerpis

Je largue les amarres

Je lève l’ancre/le camp

Je décanille

Je me fais la malle

Je mets les bouts/les cannes/les bois/les bâtons

Je mets les adjas

Je calte

Je prends la fille de l’air [sounds a bit odd with the first person actually]

Some of these (e.g. “Je file”) are only slightly colloquial and could be used in many contexts but many are obscure, very informal or have other connotations and should not be used without knowing exactly what you are doing.

Je vais

Finally,

Je vais

is not grammatically incorrect but it means literally “I go”, not “I’m leaving”. I can’t think of a context in which you could use it alone without sounding overly formal, in standard spoken French it’s only common to introduce an action that will happen in the immediate future (a bit like “I’m going to” in English):

Je vais marcher

Je vais aux toilettes

Je vais y aller

Je vais me faire mal si je continue

On a bien des choix ! Dans une direction un peu différente, construit sur la variante ancienne « to take (one’s) leave (of) » (1, 2, 3), sans doute plus littéraire et formel, mais qu’on peut aussi semble-t-il utiliser de manière usuelle, l’ancien : « prendre congé de quelqu’un » …

[…] Prendre congé de quelqu’un, aller le saluer avant de partir,
par déférence ou par amitié. Prendre congé de ses hôtes.
Permettez-moi de prendre congé de vous ou, ellipt., de prendre
congé. […]

[ Dictionnaire de l’Académie française, 9e, congé ]

Et donc un empereur peut dire un truc léger comme: « J’abandonne Byzance et prends congé de vous », [Corneille, Pulch. III, 1, au Littré]. On retiendra « je prends congé de vous ».

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Pour se souvenir de l’origine de prendre congé de qqn. (et non pas pour saluer qqn. avant un départ), un proverbe !

Pour boire de l’eau et coucher dehors on ne demande congé à personne. Pour user de ce qui ne coûte rien on n’a pas besoin d’autorisation.

[ Grand dictionnaire universel du XIXe siècle, P. Larousse, t. 4,
1869, congé ]