At the beginning, adventuring with Atreus in God of War may leave you with the impression you're running an escort mission with an AI. However as his combat ability develops Atreus can become a formidable force on the battlefield. Getting the most out of him will let you tackle God of War on higher difficulty settings and go a long way toward helping you to enjoy the gameplay. Using his abilities at strategic times and choosing good skills early is key. This guide to fighting alongside Atreus will provide some advice for doing just that. You must expect light spoilers about abilities Atreus unlocks (or else he can't be discussed in full) but I won't touch on specific plot points at all.
In battle, Atreus will handle himself. Occasionally he gets locked down, which will disable his abilities until you break him free of an enemy's hold, but for the majority of the time he is dashing about the battlefield using his small size and speed to his advantage. He will execute melee and ranged attacks automatically, though he will not use any special type of arrow when he does fire the bow and won't spam attacks. His behavior in combat is governed by three abilities and (potentially) the armor type you select early-mid game. The base damage of these attacks is really small though he can sometimes finish off an enemy. His default melee and automatic arrow attacks can be upgraded via skills, the Talon Bow, and armor upgrades that specialize him. Atreus has a few moves he'll use fairly regularly that will be discussed below.
Atreus can be directed to fire arrows with Square. He will by default target the enemy you're looking at (or who is closest to the center of the screen), but can be directed to shoot a specific spot if you hold the aim button (L2). In general it's better to not use L2 aiming with Atreus unless you intend to use a Leviathan Axe throw, for his tracking is excellent. He's a better shot than most average players could hope to be. Look toward an annoying nightmare or revenant and he will hit it with 90%+ accuracy. He rarely lets me down, but if he does he's constantly reloading to let himself take another shot.
When an enemy is vulnerable to attack (ie, not blocking) Atreus can stagger them to some degree and interrupt some attacks in progress with arrows you direct him to fire. It depends greatly on the level of the monster, specific attack being used, and the difficulty. Arrows won't do much with an enemy 2+ levels above Kratos (Atreus seems to benefit or suffer from Kratos's level). His shots, when timed correctly, can help to give you an opening for Kratos to lay in with a short Leviathan Axe combo. Use them when you're close to attacking or sprinting toward an enemy. It is often better to use R2, R1, R1, R1, R1 to help you to finish the combo while firing all available arrows during this action. Charging up an Executioner's Cleave? Nail them with a few arrows as quickly as Atreus can fire them as Kratos begins the animation. All of this applies to ranged monsters as well - Atreus can help you to land a Leviathan Axe throw without ranged retaliation by staggering a caster.
It's a huge increase to DPS to have both characters hitting the same monster when it's open to attack. I emphasize this, as earlier in the game I found myself not attacking with Atreus when Kratos was acting, and that was a mistake I'd encourage others not to make. It's not like God of War's combos are that hard to execute that you cannot tap square as you're hitting R1/R2.
Atreus' attacks can stun enemies, opening the possibility for an R3 finisher or cleave. The R3 finisher is actually the best route to go for certain encounters, because the enemies are quite beefy and you get a temporary invincibility while Kratos is doing the animation. Use Kratos' fists along with a volley of arrows to quickly build an enemy's stun meter early-game. Arrows can also be used to interrupt enemies powering up in Give Me God of War difficulty. If they are not hit in time they will both heal and level up a couple of levels, becoming much stronger. Multiple enemies doing this often leads to a game over in GMGoW.
Atreus gets three combat skills at level 1 that, when combined, dramatically change how he operates in combat. It alters his AI to be much more aggressive and intelligent. He goes from a cautious child to capable fighter with about 1500xp. You will need to advance the story a little to fully benefit from them all but they are still wonderful early-game abilities and the change is dramatic when you've collected all three. I recommend going back and forth upgrading Kratos and Atreus to balance it out.
Teamwork, Ferocity, and Acrobatics all cost 500 XP and add something to his AI that makes him better at helping you to lock down enemies (ferocity), capitalize on your combos that are initiated when he's choking them (acrobatics), distract an enemy attacking Kratos (teamwork), or provide Kratos an opening for a combo/finisher (teamwork/ferocity). I may be off on what does what in some cases but you get the point. You will want these to make the most of The Boy. I recommend you begin with Teamwork then move on to Ferocity. Atreus will have bloomed in ability by the time you reach the center of the Lake of Nine and resume The Journey toward the mountain.
As with Kratos's Leviathan Axe, skills for Atreus become available as you upgrade the Talon Bow. Magical abilities will unlock later in the game (this is not a spoiler, it's on the skills tab, though those skills are hidden). Most of Atreus' other abilities focus on increasing melee damage and firing rate. Quiver capacity is upgraded ever-so slightly with a later game upgrade (to 4). The bow doesn't require special items like the Leviathan Axe does, but you will need to wait until those more rare crafting materials come into play (such as Solid and Hardened Svartalfheim Steel). While you might be able to get a couple doing side quests they are not available in quantity until you unlock more areas of the game. With that in mind, do what you can with what you have.
I do feel that an upgrade to Atreus' bow is more important than a disposable armor piece for Kratos as it is everything to Atreus and unlocks new abilities while you cannot say the same for armor. For this reason any materials towards improving the Talon Bow should be put to use as soon as it's affordable.
God of War Tips - My Intro Page with loads of advice.
God of War Stats: What Luck, Cooldown, Strength, Defense and Runic do for Kratos
Using Stats to Make a Build
Status Effects - Permafrost, Immolation, Shock, etc.
Niflheim (End Game Content) - Increase Mist Resistance.