Versailles and Giverny

For Giverny, a tour could help you with the logistics of getting there. It is not hard on your own but train schedules can be inconsistent (there's a lot of works on the line to Giverny) and you need to connect to the shuttle in Vernon-Giverny station.
I cannot recommend a specific tour though.

For Versailles, Uber/taxi is unnecessary. There are no fewer than 3 railway lines from Paris to Versailles:
- L train out of Gare Saint Lazare
- N train out of Gare Montparnasse, and
- C train ("RER C") from several stations along the Seine (beware, the most central one at St Michel Notre Dame is closed until April and I'm unsure of the exact reopening date)

The C train gets you to the station nearest the Palace, called "Versailles Château Rive Gauche", but the one for the L train "Versailles Rive Droite* is barely further. The N train goes to "Versailles Chantiers" and that's still walkable, but buses are available too.

Is Versailles and Giverny doable in a single day?

Versailles + Giverny in a day is only doable with a tour, or your own car, but Versailles deserves most of a day, so it is not recommended.
Also, more personal opinion maybe, but I just find that Versailles has SO much more to it than Giverny. In terms of historical and artistic significance, there is no comparison for me.

"Is Versailles and Giverny doable in a single day?"
Independently? - probably not.
There are, however, day tours from Paris that combine the two as part of an all-day trip. The TripAdvisor "Things to Do" Pull Down Menu has a few that might be of interest.

I think the best answer would be, do you want to SEE them (experience them) or just be able to say you were there? If the answer is the first, then no.. As a previous responder said you really need a whole day for Versailles, and honestly the trip to Giverny and back takes most of a day. There is a delightful American art museum there to see as well.

Ok separate days it is. Now which tour? We really don't want to deal with trains. Can someone recommend a reliable tour?

Thank you for your reply, I think covering both in a single day will do injustice to Versailles

We have been to both sites, personally I think Versailles is not worth the effort. Place is always crowded and I mean jam packed. One time we did a Fat Tire Bike tour, Bike to the train station, load the bikes on the train then off to Versailles. That was fun as we had bikes to tour the massive grounds. Another time we rode the bus from Paris, dropped us off right in front of the place.
Seriously getting to either place is easy peasy. If I had to pick one it would be Giverny. The train trip to Vernon is pleasant, you can either catch a bus at the train station or rent a bike from the bar across the street and make the very sonic and relaxed 5K bike trail to the gardens. The bar used to have a great house Bordeaux for 3e too. If you will take the Metro in Paris, you can do this.
Bon Chance