I know that druids can use wild form to shapeshift into animals with wild form, but is there a way for others to shapeshift too, without multi-classing?
The Polymorph spell can allow it:
This spell transforms a creature with at least 1 hit point that you can see within range into a new form. An unwilling creature must make a Wisdom saving throw to avoid the effect. A shapechanger automatically succeeds on this saving throw.
I don't see why this could not be used on yourself, to allow you to change.
This means that anyone that has access to this spell, can transform at will (provided they have the necessary components, spell slots etc.). Classes that have access to this spell in 5e are: Bard, Druid, Sorcerer, Wizard