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.
To get Shield proficiency without multiclassing you could use the Mountain Dwarf, Hobgoblin, or Githyanki races, all of which provide you with light armor proficiency. If your DM allows the Supernatural Gift from Mythic Odysseys of Theros with feats, you could also get the Moderately Armored feat right off the bat at 1st level. Choose to get money for equipment instead of the standard equipment. Then you can buy scale mail plus a shield, for a base AC of 16 +2 Dex max = 18 AC. When you can, buy a Half Plate Armor for AC of 17 +2 Dex max = 19 AC.
Besides the options listed above, the only other way I am aware of for a Wizard to use a shield is the Tenser's Transformation spell from Xanathar's Guide to Everything. Among other things, the spell gives you:
The spell does prevent you from performing any spellcasting while the spell is in effect, but otherwise, it does allow a Wizard to use a shield as requested.