Currently the most straightforward, and the build chosen by most players when it comes to a Mage build is the Fully Offensive build. The hallmarks of the Fully Offensive Mage are: 1. The use of double Poelalas (Pet and Guest) 2. The maxing of Dexterity, Intellect, and Luck to 250 3. The use of the Vampire subrace, especially the Beast Form transformation (unlocked at Level 10) and the Mesmerize skill that can stun the enemy (unlike the Werewolf's SNarl, Mesmerize is a guaranteed stun) 4. The use of elemental armor skills, which, in my opinion, have rendered the vast majority of MP damage dealing spells a bit irrelevant 5. The use of magic weapons as your only category of weapons, because Poelalas boost Magic weapon damage to a large amount
In other words, we could say that the current definition of a Fully Offensive Mage is almost like a warrior that uses magic attacks. This could be the closest thing we see to a "Battlemage."
For armors, definitely pick up what I call the Unholy Trinity of Pre-Lean Elemental Nukes: White Knight Z for Light, Horo-Show Void Vigilante for Ice, and Taladosian Robes for Energy. They're the most powerful armor skills for their respective elements barring Kindred Strike at 200 charges. White Knight Z costs Z-Tokens, so if you're willing to get Z-Tokens it is an extremely worthy if not mandatory investment. Horo-Show and Taladosian are completely F2P.
The Bloodzerker armors are the best for Fire and Earth. They give a passive 20% buff to your weapon damage, and their trigger skill gives you the nukes for these two elements by massively boosting your weapon damage that can be buffed even further by using Chief's IronThorn, MISC items, and Arcane Amplification. It's an active trigger skill that costs a not-too-big amount of SP per turn (deducted every time you click "Attack"). Both cost an UR GGB, so if you're not willing to get UltraRare GGBs I'm afraid your best alternative would be Headless Horseman or Overlord for Fire and Cthonian Count (Earth+Ice Vampire variant).
As Primate Murder mentioned, it's worth noting the Morningstar Occult Cross (you have to buy the Magic version, the Cross comes in ranged and magic variants) has great synergy with your Vampire's Beast Form and the two Bloodzerker armors if you're able to get them. Pairing the weapon with one of the Bloodzerker armors gives you a fire or earth nuke that's even more powerful than attacking with the Berzerker swords. In fact, Terra Bloodzerker+MS Occult Cross and Blazing Bloodzerker+MS Occult Cross give you the strongest elecomp nukes for the fire and earth elements to date. Unfortunately, the downside to this is that using Bloodzerker in conjunction with the Cross can be costly on SP, as it takes 294 SP for the Level 150 version of the Cross to perform the x2 damage attack, and that is added to the SP cost of the elemental trigger of the Bloodzerker armors which is about 200 SP in their Level 150 versions. Still, with the effective management of SP using Essence Orb, Purple Rain, and stun-lock miscs like Shadowfeeder Pendant / Love Potion, you can use this combo to deal very high amounts of damage that includes being able to kill normal monsters in 1 turn. Vampire's Beast Form also synergizes well with the Morningstar Occult Cross and it gives you a light nuke, but if you can get White Knight Z then use White Knight Z as its skill has higher elecomp and thus more powerful.
For Wind, your Vampire armor covers Wind. However, if you're looking for a very strong wind nuke and are willing to spend, pick up Airenal's Lance which gives you the Lord of the Skies armor that has the strongest wind nuke in the game (SP costing). It does take up a weapon slot. For Darkness, your Vampire armor + a light weapon (including MS Occult Cross) will be enough for you to take on pretty much all Darkness monsters. However, if you're looking for an efficient but powerful (with applied buffs) Darkness nuke that does not cost SP or MP but only a small amount of HP, get Angel of Souls. Beware, the quest is long as you first have to die and collect 100 hourglasses for Death (click on the hourglass), but it's the best F2P Darkness armor in the game that's not a subrace. For your Water armor, if you don't mind the extremely ugly appearance, get WHEEL. Its skill is not MP based so it can be boosted by Chief's IronThorn. If not, your best alternative would be Hydromancer Bloodmage (which, again, costs an UR GGB). My suggestion is that once a water version of the Bloodzerker, or another FO water armor that has a skill which is boostable by CiT is released, switch to that. Of course, if you have Moglord Tortress, there is no reason you should be using another water armor as Moglord's skill is the #1 water nuke in the game.
I disagree with using Lumenomancer Bloodmage over White Knight Z and using Cryomancer Bloodmage over Horo-Show Void Vigilante as the latter options have more powerful skills. For weapons, currently I'd recommend setting your no-drop element to either Wind or Energy unless you have rares for those elements. Kindred's not effective or efficient as a regular questing armor, so perhaps use Angel of Souls or a Vampire variant in place of Kindred for regular questing to make up for any element where you don't have a nuke or a cheap, efficient spammable skill (Angel of Souls). You could even use Whispering Raiment for its Initiative boost against monsters with high Luck, or Wizard Robes (Generalist Robes) if you want to cast spells.
Your weapon inventory should look something like this:
-Staff of Awe (upgrade to Staff of Awe!!!! through Golden Giftboxes if you can)
-Terra Bloodzerker's Sword (Earth, rare GGB)
-Bloodzerker's Blazing Sword (Fire, rare GGB)
-Ultra Omni Knight Blade (Water / Ice compression, use Hydromancer and / or Cryomancer Bloodblades if you have them) or the Kindred set sword or Cthulass or the Celtic set weapon
-Agony's Embrace (Darkness, UltraRare GGB), if not, I'm afraid your only option is Paragon Ice Blade and that means you'll have to be without a water weapon, but Water / Ice in my opinion is less of a loss to be missing one of these elements than it is to be missing one of the other elements
-Zealot's Wrath / Lumenomancer Bloodblade for Light, if not, Morningstar Occult Cross+Vampire's Beast Form or Elahi Crook of Osiris (from the Osiris set quest)
-The Fujin set dagger for Wind (Kazemai Athame) or Lord of the Skies as your main source of wind damage (through Airenal's Lance), the Fujin set dagger is OK but not ideal, if a FO magic weapon for Wind is released in the future, you should probably switch to that
-The last slot can either be used for a spellboosting weapon like Sila's Staff / Tsunami Pen (if you choose Hydromancer Bloodmage) / Arctic Athame (if you choose Cryomancer Bloodmage over HSVV) or for your water / ice weapon if you've got the Bloodblades
*A weapon worth noting is Void Spear of War from Rare GGBs. It's currently one of the few sources of Void weapon damage in the game and has a Magic form that can activated by pressing on the weapon. Pick it up if you can. Before that, use Archmage Research in one of your spell slots. Void Spear of War can be substituted in place of one of your weapons when you're going to fight monsters with very high resistances to the 8 main elements.
Shields: Use a Compression shield to fit in CiT, the rest.... I'm not an expert so you should ask someone else. Most of the time you'll be relying on CiT as a damage booster but this does not mean elemental shields are useless. Your ice shield should be Pies, as it gives a CHA boost that increases Vampire's Mesmerize skill's chance of succeeding. This does not mean you should train CHA as you shouldn't. a FO mage with Peolalas does not need CHA because you won't use pets / guests to attack.
Spells: Purple Rain, Moonwalker's Grace, Destruction Burst, Arms of the Dragonguard (UR GGB, get it if you can because it gives pretty much guaranteed Lucky Strike chance for 2 turns), Arcane Amplification (costs Z-Tokens but get it if you can, INT boost by 102 at Level 150 for 3 turns), the rest should be filled with utility>damage spells which you should ask someone else what the best choices are. Mages these days rely far less on MP damage spells than in days past, as armors and weapons have taken over as the main sources of damage.
MISC Items: Shadowfeeder Pendant and Essence Orb are mandatory in this meta. The rest of your slots could be filled with miscs like Rejuvenating Necklace (heals MP based on damage done), Bell Shell (heals HP based on damage done), Cyclop's Eye, Blood Contract (costs Z-Tokens but it's a general damage booster for armor skills) / the Warlic misc from the UR GGBs (INT boost), the Osiris misc (INT and BTH boost, light resistance), use Love Potion if you have it as it gives an extra turn of stun, Legendary Shadow Crystal V1 (but only on Naga Day), and Taladosian Pendant (increases chances of Vampire's Mesmerize succeeding).