Train from Arles to Vernon

Best as in fastest? cheapest? most scenic? easiest? It all depends on your priorities, but normally travelers just indicate their origin and destination and choose from among the options presented based on time and cost.

It's a pretty safe bet in France that if you're traveling between towns in different regions, you're going to need to go through Paris. You can go to the Deutsche Bahn website (which is quite easy to use), enter Arles and Vernon, and it will present the logical rail options. Click on "Show details" to see where you will be changing trains. If you want to see what other stops the train makes, click on "Show intermediate stops."

Some of your possible departures involve two changes in Paris, not just one (plus other changes elsewhere). Some of those Paris transfers indicate that you just walk to a different area (possibly under the same roof) to find your platform. The others are the ones to take note of, because they mean hopping on the Metro or taxiing to a different rail station. It's a manageable thing to do if you're traveling light; it's not so much fun if you have heavy luggage.

To check fares you'll need to use the French rail website, SNCF, or perhaps trainline.eu.

The TGV components of your trip will probably increase in price as your travel date approaches. When tickets go on sale, there are some promotional fares available. As other passengers snap up those good deals, you are left only with higher-priced tickets.

In checking the Bahn site, I see that the train comes into Paris - Gare de Lyon, and I have 50 minutes to transfer to Paris St. Lazare to continue on to Vernon. Is 50 minutes enough time? What are my options to transfer between the two train stations? Is Metro easy to use? Are taxis safe?

"50 minutes to transfer to Paris St. Lazare to continue on to Vernon. Is 50 minutes enough time?" - Yes, you should do it in about 30. But, as a newbie who has never crossed Paris before, might take a wrong turning or get nervous, I would recommend you investigate an earlier train into Gare de Lyon or a later out of Gare St Lazare. Say 90 minutes between trains.

"What are my options to transfer between the two train stations? Is Metro easy to use? Are taxis safe?"

Yes, métro or taxi. Métro map here: https://www.ratp.fr/sites/default/files/plans-lignes/Plans-essentiels/Plan-Metro.pdf
Line 14 (purple) all the way. You would have to buy a métro ticket (€1.90).
Taxi's are safe, but a lot more expensive. Not necessarily faster. Again a taxi may be the less stressful way for a newbie.

Nice things about the No 14 line.
1. Very few stops. Just 3 intermediate stations between Gare de Lyon and Gare St Lazare. So it is fast.
2. Elevator and/or escalator access to all the platforms at the stations, so easy to navigate with luggage. That is because it is the newest Metro line and deep underground, so stairs are not practical except for emergency situations.

Is it possible to buy Metro tickets online in advance so that I at least have that and just need to get to the Purple Line?

I have heard, but never experienced, that they may sell 10-packs of tickets in the bar car of TGV's to Paris, but they would cost like 15 EUR for the single 1.90 EUR ticket you need. They sell tickets in vending machines and there will be a lot at Gare de Lyon, as well as the manned ticket windows. Shouldn't take long.

Will you be returning to Paris from Giverny? If you are going to Vernon I guess that's what you are doing.

If so, any unused "t+" tickets you have bought as part of a carnet (10 tickets bought at once for the Métro and buses in Paris) will still be valid and could be used. They never expire.