What ways are there for a wizard in D&D 5e to use a shield?
The only option I could think of was multiclassing which does not seem worth it.
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The only ways for a wizard to gain shield proficiency are either multiclassing, or the Moderately Armored feat. Since Moderately Armored requires you to be proficient in light armor, most wizards will need either a multiclass or another feat (Lightly Armored).
Note that you can use a shield without proficiency - you just won't be able to cast spells, and you'll have disadvantage on any attack roll, saving throw, or ability check that involves Strength or Dexterity.
If you select hobgoblin as your race you start with proficiency in light armor and may take the moderately armored feat giving you a proficiency with both medium armor and shields allowing for an ac of up to 19 (24 with the shield spell)