There is an excellent series on youtube called "A friendly introduction to group theory"[1] which takes in my view a very intuitive approach of starting with symmetry groups. There's also "Group theory and the Rubik's Cube"[2] which teaches group theory starting with the symmetries of the Rubik's Cube. I personally think starting with symmetry groups and later on showing (via Cayley's theorem or whatever) that these are isomorphic to integers modulo n or general cyclic groups is the way to go to build intuition.

[1] https://www.youtube.com/watch?v=4n1BhWzdVsU

[2] https://people.math.harvard.edu/~jjchen/docs/Group%20Theory%...