As you can see the part around the handle is pulling really bad. Was it not enough divisions from the very beginning? How do I fix this problem without starting all over? Thanks a lot!
The main problem is that the handle is extruded from the middle of a face instead of from a proper "edge looped" one. Your model ends up with non quads just above the handle and subdiv modifier does not like that.... At this point you might be better off starting again from a cylinder with enough subdivisions for one face to match the size of the handle. Check out the difference in topology between the two models below. The left one started from a 10-face cylinder, the right one from a 20-face cylinder:
hi as previously stated by other replys you have n-gons in your geo, while not generally bad, it's hard to predict how they will subdivide on curved surfaces, but that is not causing your "pull" problem. Your "pull" problem is caused by the 2 horizontal reinforce loops near each handle, they "amplify" the pull effect.
This is what you did:
If you are in a hurry here is a cheap solution, still you would have those n-gons and from the side you have no control on the transition:
This is my suggested solution:
Here is how it looks:
This happens, because your model has n-gons near the handle.
N-gons are problematic for the Subdivision modifier, and lead to unpredictable results.
Use closed edge loops and quads instead and Subdivision will work better.
Add loopcuts around the whole mug and use To Sphere function, to make it round again.