First off you probably can not put the bathroom ANYWHERE so give us all an idea of where it will go then everything else can be placed. Draw where the rough ins are and hopefully you have rough ins because if not that is going to be a really big expense.

1st, you need to show the exact dimensions of everything, show column locations, where the main beam is located, and where the main ductwork runs, and

What is the ceiling height?

What would you like to have in a gym area, what type of equipment?

In the bathroom, do you need a shower in it or not?

You provided lots of things that you need, but you left out important details about what is already there and their exact locations and distances from each other. You need all that to get an accurate layout you can work with and come up with a plan you can start framing walls, etc, knowing everything will fit as you like.

If you need help, please drop me your email in a message on my profile page, and I will be more than happy to go over it with you.

Good luck

The bathroom will need to go somewhere near the back corner of the basement because of where the pumps are.

What type of pump is there? The only pump you show is a sump pump near the water heater.

Has an ejector pump (used to pump out the sewage) already been pre-installed under the slab where you are showing?

By the way, where is the sewer line that comes in from the street? And if that's the case, does the sewer line come into the house through the foundation wall or the slab?

Pictures of the existing space, room dimensions, and understanding of where your existing plumbing lines are would be helpful here.

Here’s an updated drawing of the basement with dimensions and plumbing and windows