Are your flights already purchased? Have you been to London before? I would not include it for just one day unless it is a matter of getting an affordable flight. I think splitting your time between Paris and Rome would be more reasonable. But what is it that you (and your fiance) want from the trip?

I have a similar view but I'd drop Rome (apart from the fountain) hardly romantic. That leaves
Paris
Nice
French Riviera
Cinque Terre

Not a bad 15 day trip (Could fly by from Milan perhaps)
On the other hand if you guys love trains you have a good trip already planned.

You really have 13 days here, and if you include both Rome and Paris I suggest a max of 3 destinations.

Arrive Paris---5 nites
Train to Nice---3 nites
Fly to Rome---5 nites

That makes a great honeymoon.

"... we still want to experience a lot. "

... then you need to travel less and see more

This is an unholy mess. You have one day in London because your arrival date is all jetlag and recovery with nearly no touring. You have 2.5 days in Paris, a city that deserves about 2x that for a minimum stay. You have mere hours in Cinque Terre.

And what's the 6k, your budget? Good luck with that with all these itinerary stops that require PAYING MONEY FOR TRANSPORT. Your flights to and from the US will be 2500, that leaves 3500 for the rest and you'll need more than 1/2 that for accommodation ($150/night for 13 nights = $1950 and that's NOT top-end facilities by any means) unless you want to tell your kids about how you spent your honeymoon in dumps.

Review and revise.

I'm afraid that you really need to cut back on destinations - if you want to actually see anything. And your budget is definitely budget (perhaps more modest that you are prepared to deal with).

The more stops you make the more expensive the vacation (more time AND money spent on trains). And since you are going in high season you can't count on hotel bargains - unless you are planning on staying in hostels. Remember that besides meals you have to allow for sightseeing (not cheap - have a look at the entrance to a few major sights) as well as local transit and having a drink in a cafe or an ice cream on a hot afternoon.

In the amount of time you have i would stay in 3 (three) hotels and give yourself time to relax a little - and perhaps do a day trip from one or more of the cities.

I count 12.5 days on the ground. I would pick 3 destinations max. The easiest way to keep a smaller budget is to move around less - plus you will have more time to actually enjoy and see the places you are visiting.

First thing I would do I drop all one day stops...London for 1 day doesn't make a lot of sense and adds a lot to you budget.

I would either do something like London, Paris, Rome - flying into London and out of Rome or stick to one country and do France or Italy.

In cities look at renting apartments so you can eat in to save money.

I agree with all of the above. To save time (and likely money) fly into Paris and out of Rome. Visit Paris, Nice and Rome. Cinque Terre is definitely not worth it for the part of a day you will be there.
$6K may work if you get a good price on your flights but will require that you do a lot of work to keep accommodation costs down. You can get help on that here once you decide on an itinerary.

Consider traveling in mid June or as late in September as you can to avoid the biggest crowds and the hottest weather.

Good luck and congratulations.

"Also, what is the best way to travel among the countries: train, car, or plane?"

A Star Trek transporter would be best.
Good luck!

If I cut out Italy all together and focus on London, Paris, and the French Riviera, what would be the best itinerary?

I can't seem to figure out where I would fly out of. What would be the cheapest and easiest airport to fly out of from the french riviera?

I really appreciate everyone's input.

No one can tell you what the cheapest airport is - there's no such thing. You have to check individual flights and make your own determination.

Your choices are really Nice or Marseilles.

You actually only have 12 travel days. I'd spend 3 days in London, 5 days in Paris, and 4 on the Côte d'Azur.