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As for the rest of the subclasses, it depends on the role you want your wizard to play. If he's going to hang back and be a backline blaster/aoe caster then shock arcanist is by far the best. In addition to casting all spells one level higher for free, at higher levels they get things like arcane fury and arcane shock that add things like int bonuses to damage and higher average damage dice. Shock arcanists deal in maximum damage from their spells
If you want your mage to fight up front then court mage is the best. Court mage is arguably the best subclass for mage. You gain proficiency with a shield, the protection fighting style, big temp hit point shields. A bunch of other stuff I'm forgetting at the moment because I mostly play shock arcanist.
But yea, that's basically the deal. Shock arcanist for range/backline, and court mage for front line.
I can't think of any really good reason to ever want to play a lore mage, unless you're just really into crafting and scribing scrolls and so forth. I think at higher levels they can memorize more spells per level than either shock arcanist or court mage, but you have to reach those levels first to get that. Shock arcanist and court mage begin paying dividends much earlier on.
Studded Leather +2 will provide more AC than Mage Armor and the armor may also provide an additional benefit like save bonus.
Having access to druid spells as a wizard is more of a novelty thing rather than really useful imo since these additional spells still compete for the same spell slots as your 'regular' spells.
As far as the Shock Arcanst, it's good, but I largely found that I didn't really need its damage boosts. I never even used Arane Shock due to restrain and damage it inflicts on the wizard using it.
I actually had not intended, originally, to play a Greenmage, as it felt like the concept was more suitable for elves and I had decided to create a human mage- just for variety, as the rest of the party were demihumans, But somewhere along the way, I read that the utility offered by the Greenmage was helpful and after seeing the weapon and armor proficiencies, decided to give the subclass a try.
Given that my mage has Dex 16 and Int 19, I think that her cantrips outclass the bow, even at level 2, especially as she is not an elf and won't gain longbow proficiency. To confirm, the Int bonus adds to both the bonus to hit and the damage of the cantrip? At least it seemed that way.
Again, I had forgotten- when I made this choice- that cantrips in 5E are unlimited. Firebolts and Chill Touch seem to have a lot of utility, which a bow does not grant (unless there are elemental arrows in the game). I guess I'll just see how it actually plays out the rest of the game- I'm sure I can make it work even if it does not feel fitting for the way I built my character.
Your ranged attack is also higher then spell attack because archery style gives + 2.
Later, when you can craft magical bows that add bonus dies to damage, such as the Medusa Shortbow, the shortbow could be competitive with your cantrips before 11th level. I believe the Medusa Shortbows adds 1d8 poison damage on a hit. Of course, one can always just use poison arrows which are commonly found loot,
If you have a ranger or druid the sub-class becomes redundant.
(You don't need 2 people who can cast pass without trace or goodberry.)
Mage Armor ties up a precious spell slot and if you have a travel encounter you have to waste your first action casting it. And if you were surprised you have a dangerous first round indeed. Light armor seems preferable to me, especially once you have +1 or +2 light armors available.
Greenmage also has the largest spell list of any Wizard. At low levels, Entangle, Faerie Fire, Pass Without Trace and Goodberry are all excellent spells. If you are into that sort of thing, the Greenmage gets a monster summoning spell at 5th level (Conjure Animal) where other Wizards don't get one until 7th level (Conjure Minor Elementals). Contagion is a great single target debuff that does not require concentration.
I will spit an fireball out my *** once in a while but the focus is sniper archer with cc arrows.
Int bonus does not get added to cantrip damage so until you reach level 5 and your cantrips start using two dice, the bow is usually better damage-wise. And later in the game there are some decent bows with additional effects that are worth looking at.
As far as I'm concerned, Greenmage is the best Wizard subclass. I've gone Shock Arcanist a couple times and never noticed the extra damage to be something that turns the tide in any particular fight.
Edit: I'll also throw in that it's definitely worth taking the Uncanny Accuracy feat for any Wizard. It does work with ranged spell attacks.
1) Unless you are playing a draconic sorcerer or a warlock (newest DLC) cantrips do NOT get bonus damage from your casting stat, bows do get bonus damage from DEX.
2) The attack bonus for for archery weapon style is +2, which is a HUGE bonus as to hit bonuses are hard to come by.
3) You can get bows that add +dX damage in elemental form and you can craft arrows that add +1d6 elemental damage, which means you could have, say, a stormbow doing 1d8+1d10(electric)+1 damage with arrows of acid doing +1d6 damage for a total of 3-24 + DEX bonus damage.
4) while a longbow is superior to a shortbow, its only +1 damage better on the average so unless you are an elf it is probably not worth burning a feat to get longbow proficiency.
This means that if your dex is anywhere near your casting stat, a bow or other ranged weapon will generally be superior to a cantrip up to level 5 and is likely to be a good choice up to L11 when cantrips go up to 3dX. For a greenmage unless your INT is 6 points or more higher than your dex, it will always be a better choice up to L5 and usually will be better up to L11. After that it STILL might be a better choice, depending upon what bow/arrows you have available.
As mentioned by previous posters a greenmage is meant to be a sort of cheap multiclass mage/druid, so a real druid will always be better at the druid stuff and a ranger will always be a better archer and both can cover any food requirements with greenberry. So if you have either of those in your party your best off with a shock arcanist as they have HUGE amounts of burst damage and with food requirements covered you have plenty of opportunities to rest between fights in both the base campaign and the Lost Valley.
That mostly comes down to lining things up well, but I have landed some big mob wipes with Fireball and Cone of Cold with it.
EDIT: Looking at what people are saying ignore the weapons. You have cantrips which aren't the main focus either. You want to change the tide of combat with spells. You shouldn't be trying to compete with other classes in weapon strikes without stats for it, fighting style, or extra attack, etc.