When determining your character's ability score, increase your Dexterity by 2. Alternatively, if allowed by the DM, increase any one ability score of your choice by 2, or 2 different ability scores of your choice by 1 each

Gain proficiency in Athletics or , if allowed by your DM , another skill of your choice

You character can speak, read & write in Common fluently!

Choose between learning 1 language (not Common) or gaining proficiency in 1 tool or weapon of your choice

If at least 3 of your 4 hands are free, then your climb speed is equal to your walk speed, whatever that may be (your base walk speed = 30ft)

A character of this race cannot use the Extra Attack feature. When offered the Extra Attack feature by whatever source (for example, when reaching the 5th Barbarian level) then you may choose a Feat (or similar feature) instead. Any feat must be approved by the DM first. You cant take a Feat multiple times this way, meaning you can choose a feat to take only the 1st time you would normally get the Extra Attack feature (for example, if you are a 10th level character with 5 levels in the Barbarian class then 5 levels in the Fighter class, then you would be able to take a feat when you reached the 5th level as a Barbarian but not again when you reached your 5th Fighter level)

Your character can never take the feat Dual Wielder

Your character has an extra bonus action which can be used only when applicable for activities involving your free hands as detailed below

If you have a free hand, you can use a bonus action to perform anything a 2-armed creature (like a human) would be able to do as a bonus action. If you have a free hand but you're already using your bonus action fo something other than casting (that something may very well be using another free arm for a different activity) , you may use your 2nd bonus action to use an object / catch or hide an item / grapple a creature / give advantage to yourself in an attempt to disarm a creature / improvise something (DM's call) etc

The free hand cannot be used for casting at the same time as performing other actions. Casting still requires an action unless specified otherwise. Also, casting a spell with a casting time of bonus action, uses all bonus actions to do so, both your normal one and the one provided by your race, even if you have an extra free hand to use it with (meaning you cant cast a spell as a bonus action & attack or use your free hand as a bonus action and vice verca)

Using all 4 of your arms with appropriate finesse takes a toll on your mental focus. After having used all 4 of your arms for anything you cant cast magic for 1 minute & vice verca, casting magic makes one of your arms lose significant motor skills, only able to be used for spell somatic components, for 1 minute (that effectively means that if your character wants to use magic, 1 of your hands must always remain free, dedicated to that use)

Having 4 arms is beneficial, but also cumbersome in terms of weight, coordination, energy expenditure when using them etc
In that sense, when using them in combat, you cant add any ability modifier as a damage bonus to any of your attacks made with them. This includes physical attacks, so melee (unarmed or weapon) or ranged attacks, although it is still possible for you to increase the damage of your attacks in different ways
Also, this doesnt affect using your spell casting ability modifier, if applicable, to your spell attacks
Finally, you can boost the damage of your attacks temporarily. When attacking with your arms as detailed above, you can simply choose to use your modifiers as a damage bonus until the start of your next turn. You can use this feature a number of times equal to your proficiency bonus & you regain all expended uses when you complete a long rest

You need specially made armor that fits you (cost determined by the DM) . It takes you double time to don or doff armor, except shields. Depending on your clothing, your arms being visible may induce negative reactions to creatures that notice them, or simply attract their attention, which could translate to penalties to your stealth or even your charisma, usually in the form of disadvantage.
Of course there may be no negative effects whatsoever, this is ultimately for the DM to decide

Wielding multiple shields offers no benefit, but quadrumanals can use a special kind of shields, called Greatshields, that offer +3 to one's AC instead of the usual +2 (yes I went for a new item, why not?) . These Greatshields, are usually bigger in size & bulkier than normal shields, and have the heavy & two-handed properties. The two-handed property in this shield's case means that while it can be carried with one arm/hand, it requires two hands to be wielded effectively & gain its AC bonus
If a character wants to use holy magic with all 4 of their hands full, the only way to do so is by using this greatshield, with their holy symbol emblazoned on it in an obvious & visible way, just like they would with a normal shield. Casting this way is an exception to the race's limitations when it comes to magic & it is only possible because the 2 arms wielding the greatshield are both performing a simple and similar enough function as to not require as much mental focus. However, using a normal shield in one hand as a holy focus, would still require a free hand for casting per the race's magic limitations as already described

You can use all 3 of your arms in fighting with weapons, in this manner
-> 1 two handed weapon (attack as action) & 1 light weapon (attack as bonus action)
-> 3 non heavy weapons (2 attacks as action & 1 attack as bonus action)
The 4th arm can remain free or be used to carry a shield or a spellcasting focus or some other item

You can use all 4 of your arms when fighting, in this manner
-> 1 two-handed weapon (attack as action) & 1 two-handed weapon (attack as bonus action)
-> 4 light weapons (2 attacks as action & 2 attacks as bonus action)