Train tickets to Bath and Oxford: which is the best way to go?

Your choice is between GWR and SWT to Bath (although the latter is a much slower route via Salisbury with few direct trains) and between GWR (Paddington) and Chiltern (Marylebone) to Oxford. From Willesden Green the latter might take a little less time overall.

I cannot imagine what jokes you may have heard about National Rail. It is simply a website that finds and displays routes and fares for all the train lines. You pick one you like, and it sends you to the correct site for the train company to buy the tickets. You do not need to figure out the name of the company (GWR, Virgin, etc.) in advance. We always start there for our UK train travel.

http://www.nationalrail.co.uk/

You actually have left it rather late for travel to Bath on March 8. The best Advance fares for early morning travel are gone, and the price is now £42.50-£85 to leave London before 9:20 am. At 9:20 you can take a "slow" train (2.5 hours) for £14, or you can wait until 10am to depart and take the "fast" train (1.5 hours) for £17----and arrive in Bath sooner.

Advance tickets for the return in late afternoon on March 9 are still a good price---£22.50.

"I cannot imagine what jokes you may have heard about National Rail."

I wonder if the OP is thinking of the defunct organisation knows years ago as British Rail:
https://en.wikipedia.org/wiki/British_Rail
Or maybe "Britrail", that foreign owned company that sells passes.

Yes, I hope that is the case, and she is confusing National Rail with something else.