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The first word that jumped to my mind is the adjective belligerent.

Some published definitions include:

• inclined or eager to fight; hostile or aggressive.




• Aggressively hostile, eager to fight; acting violently towards others.




• hostile and aggressive: a bull-necked, belligerent old man.

Such a person is a hothead or a bruiser, or has a chip on his/her shoulder.

According to Vocabulary.com,

A hothead is someone who's suddenly and easily angered or agitated.
It's usually a hothead who starts a riot or turns a peaceful protest
violent.

^{Source: https://www.vocabulary.com/dictionary/hothead}

According to CollinsDictionary.com,

A bruiser is someone who is tough, strong, and aggressive, and enjoys
a fight or argument.

^{Source: https://www.collinsdictionary.com/dictionary/english/bruiser}

Pugnacious fits:

Eager or quick to argue, quarrel, or fight.
_{Oxford Dictionaries}

You can also check its synonyms. Merriam Webster has a good comparison of them.

The adjective quarrelsome is a good word to describe someone who likes to argue with other people. Many times, they do that just for the heck of it. A quarrel is a fight, but not one that's physical. The following is the definition of quarrelsome as defined by the Cambridge Dictionary:

A quarrelsome person repeatedly argues with other people.

Another possible answer, though more general, is "instigator."

Definition: "a person who brings about or initiates something." (Google)

You will still have to use the word "fight" (or a synonym of it):

He really likes instigating fights.

I still think belligerent is a better answer, but using "instigator" specifically focuses on the fact that the person in question likes to initiate the fights, as opposed to someone who has a high temper and reacts to instigations violently.

You may want to use the word troublemaker.

A person who consciously or unconsciously causes trouble (merriam-webster).

It's flexible enough to be used for physical or verbal situations. The word is also free from profanities and is often used in professional settings.

Consider the words below:

• 
  Scrapper (informal) - a fighter or aggressive competitor, especially one always ready or eager for a fight, argument, or
  contest (source).


• 
  Fighter - a person who physically fights another person (note that the word is often used to talk about professional boxers and just strong people who don't give up easily, but in the right context the word may be what you are looking for).

By the way, a shit stirrer is a person who takes pleasure in causing trouble or discord. It's not specifically someone who loves picking fights. They may like that too, though.

belligerent, contentious, aggressive, truculent, combative, pugnacious, etc. depending on the exact context. I agree with Laurel that M-W has good guides for choosing the right word - you can see one such guide here under “choose the right synonym”.

If you're talking about a non-physical fight, "rabble-rouser" is a clean alternative to "shit-stirrer".

Instead of saying

• He's a shit-stirrer.

You can say

• He's a fire-eater. (synonym of belligerent in this case)

From the American Heritage Dictionary:

fire-eater:

2. A belligerent person or a militant partisan.

Other synonyms: bellicose, pugnacious, combative, belligerent.

These adjectives mean having or showing an eagerness to fight.

More referring to willingness/tending to get into an argument: antagonistic, contentious, quarrelsome, argumentative...

Regarding idioms, you can simply say

• He's always looking to pick a fight / quarrel. (in itself an idiom, for both arguments and physical fights)


• He's always asking for a fight / trouble.


• He's always crusing for a bruising. (slang)


• He's always spoiling for a fight.


• He's always arguing for the sake of arguing / the argument.

Slightly more specific, but appliable to general cases:

• He's always arguing the toss. (chiefly British, arguing over something already decided)


• He has a chip on his shoulder. (showing resentfulness)


• He always ends up cutting up rough. (chiefly British, suddenly becomes angry and violent)

Another option which no one else seems to have mentioned: Combative, defined as:

ready or eager to fight; pugnacious

I like fractious — it originally referred one inclined to make “fraction” (discord), now it means “irritable; argumentative; quarrelsome”.

“...in his present fractious mood, she dared whisper no observations, nor ask of him any information.”

— Charlotte Brontë, Jane Eyre

If it's someone who picks fights they know they'll win, they're a bully. This still works when the fights are purely verbal.

Someone who simply likes to fight, and may not necessarily bear any ill will towards their opponent, is a brawler. This only applies to physical violence.

If they're prone to picking verbal fights, you might describe them as argumentative (adjective).

Someone who is prone to sudden and unpredictable violence may colloquially be described as a psycho, nutter, or variations on the theme of mental instability.