Wizards do not start with armor in DnD 5e, nor do they start with any armor proficiency. Wizards don’t typically wear armor, relying instead on spells like Mage Armor and Shield to protect them from attacks. Additionally, Wizards can increase their AC (Armor Class) by increasing their Dexterity modifier, since a character’s base AC is 10 + Dexterity modifier.
While Wizards can wear armor they aren’t proficient in, it is not advised. When a character wears armor they don’t have proficiency with, they are unable to cast spells (Player’s Handbook, page 144).
If a Wizard wishes to gain proficiency in armor, they can multiclass into a class that gets armor proficiency or pick up a feat (Lightly Armored). Most commonly, a Wizard will dip into Artificer, Cleric, Druid, Fighter, Paladin, or Ranger, as these all get medium armor proficiency from multiclassing into them.
Artificers are a good choice because they also use Intelligence as their spellcasting ability, but Clerics and Druids can also work well, as can dipping into Fighter for Action Surge.