I'd like to install a second 200 AMP panel in my basement (about 25 feet from the existing one), and then as I am rewiring the old, non-grounded wires, run the new wires to the new panel.

Is this feasible, or do I just have to get an electrician to move the entire panel?

Well, let's see:

You need to talk to the power company to see if they will allow you to move the panel. Then you have to make arrangements with the power company to agree to move the Service Entrance equipment to match the new locations. This work has to be done by the power company as it is their equipment.

Then there's the matter of the permits and whether your AHJ will allow you to do this work (any mistake could kill or burn the home down),

Once you get past there, you now have to the costs associated with purchasing the required conductors and laying the circuits according to the AHJ level of NEC code required.

As Stan has said before, your best tool would be the telephone to call several electricians for bids.

If you settle on an electrical firm, you want them to get the permits since that would require them to correct mistakes they made during the project.

There is no way around the requirements of the power company or the AHJ.

As Stan has said before, your best tool would be the telephone to call several electricians for bids.
Hopefully they are charging for estimates.

:)