Draw an isometric of your proposal please.

The diagram is confusing and distances are required to give an answer. So, yes, an isometric drawing is needed. I would note a few things though. 1) check that your state an local codes allow the home owner to do such changes; 2) get the required permits and inspections and 3) even if codes allow you to do the work maybe think about using a plumber.

Below are the photos of the plumbing layout and the diagram. The red circle indicates the shower drain that I am thinking about moving across the joist and was inquiring what's the best way to do it?

My town allows homeowner to do own plumbing repairs and I am working on the permits.