Is it acceptable or required for bathroom lights to be connected to the GFCI outlet such that if the GFCI trips the lights go out?

It is acceptable for the lights to be wired this way but not required. The GFCI is only required to protect the actual receptacle(s) in the bathroom. ----

If you do not understand the work well enough to accomplish it yourself properly and safely, don't try it. Consult a professional electrician, as they are proficient enough to do it properly and safely. When working on electrical circuits and equipment, make sure to de-energize the circuit you will be working on. Then test the circuit with a definitive means to make sure it is off (multimeter with metal tipped leads, voltage tester with metal tipped leads, etc., not a non-contact tester, which is non-definitive.)