“What activities can you not do after a hip replacement?”

The answer depends o your level of activity, Before your surgery. You should be able to do essentially every thing you did before surgery.
Running is likely to be out but swimming & biking will both be possible. In fact you will have a better long term result if you get into and maintain good muscle tone before & after your hip replacement!

It depends on the approach that the surgeon uses, for example, for posterior approach, no hip internal rotation/bending down for several weeks.