Best Rail Pass for 2 weeks in 3 countries...

connecting flights between several European cities. . . . . My train travel will get me from airports to city centers and around each city that I visit.

You said it yourself, if you are flying between cities, and your only train travel is within cities, that IS NOT a lot of train travel, a rail pass is not a good idea.
Rail passes do not include city travel, not Paris métro nor London tube. If you are doing day trips from Florence (for example Pisa), these are local trains which you can buy tickets for on the day.
Again in London, buy an Oyster Card, and fill it with some cash. This will cover all trains and buses in London at a discounted fare.
"and England", what other places are you visiting? Are you staying overnight anywhere? Tickets to the usual destinations (Bath, York, etc.) are discounted if bought in advance.

Have you bought your air tickets yet? For travel in "Early July", just 6 weeks away, I hope you have.
If you hadn't, I would have recommended flying into Italy (Milan or Rome), flying to Paris, then train to London and finally flying out of London. Paris --> Florence --> London is doubling back on yourself.

If sticking to big cities and mostly flying -- NO Pass. Because the passes generally don't cover local, in city, travel. Side trips around Florence would be by Regionale trains that are very, very cheap. You probably not on the right track.

All brilliant advice. I have all my flights booked. In England I'm going to Exeter and Manchester, besides London area.
I'll definitely get an oyster pass in London.

What about getting a one or two day pass for Paris or Florence?

No, if you want to go somewhere you go to the train stations and buy a ticket.

"I'll definitely get an oyster pass in London."

Note that the Oyster Card is not a "pass." It is a card which can hold various kinds of ways of paying for fares for transit in the London area. Since you will be in London itself for only a few days, what you want to do is load the card with £20 (the card itself costs £5). Then, you tap your card to the reader on entering the tube and exiting the tube ("tap in" and "tap out"); you only tap on entering buses, but not on exiting. The Oyster computer keeps track of your rides; you will never be charged more than the day rate for the zones you travel through, but will be charged less if your travel adds up to less. If you run out of credit, you can add more at Oyster machines in stations (there are usually staff to help). When you leave, you can either keep the Oyster for you or someone else to use on a future visit (it's fully transferable), or you can get back both the remaining value and the £5 cost of the card itself, provided you always used the same method to buy, load, and reload it (cash or the same credit card).

"What about getting a one or two day pass for Paris or Florence?"

For Florence, the center is walkable, and the only local bus most people use is the one up to Fiesole (a very worthwhile trip). Places besides Fiesole (such as Pisa, Lucca, Siena, San Gimignano, etc) are on regional trains or buses that are not part of the local Florence bus system. So, a "bus pass" for Florence is a waste.

For Paris, you can buy a Mobilis ticket, but it takes 5 rides a day for this to break even. So, you may do better just buying a carnet of 10 regular metro tickets; these tickets can be shared with others if you don't use them all. Paris is changing its ticketing system (there will be a card similar to the Oyster card which will hold single tickets and day passes; right now, the Navigo card is only for passes of one week and longer). I don't know when the new system goes into effect, or what will happen with the old tickets when they change to the new system.

For your travel in England to Exeter and Manchester, the least expensive option by far is to buy Advance tickets (with a capital A) ahead of time for a specific train. They are generally offered at the lowest cost 11-12 weeks ahead of the date of travel. They are for a specific train and are non-refundable, but you can save a lot.

For example, yesterday we traveled to the Dorset Coast on a train to Wool. Had I bought the tickets at the station, we would have paid £63 each. But our Advance tickets, purchased 11 weeks ago, cost £13.50 each. So we saved a total of £200 on our round-trip journey.

In England I'm going to Exeter and Manchester, besides London area.

in your original post you say:

finally London and England (6 days).

Manchester is 2h05 each way, Exeter is 2h15. Adding the time to get from your hotel in London to your departure station, I would say that is too far for a day trip. And you only have 6 days in London, not enough for 2 overnights plus London.
Exeter and Manchester are also strange choices. Exeter is a nice old town, worth ½ day. It is not "first division" though, more usual and closer choices for old cities are Winchester, Salisbury, Canterbury, Bath, Oxford, Cambridge and others. Manchester is mostly an industrial city.
If you are going in early July, you need to finalise your plans and buy advance tickets for your trips from London now (not tomorrow).

P.S. Delete the word "pass" from your vocabulary. You buy "tickets" from A to B, for each trip. The Oyster Card is an electronic wallet that allows you to get tickets at the lowest price. You charge it with some money, and use it to get through the turnstiles on trains, tube and buses in London, and it charges a deeply discounted fare.

Such excellent advice. Thanks for sharing your experience, correct terminology and the urgency of advance booking.

I will be in London for 3 nights and then take the train to Exeter where a friend is picking me up. We will drive to his home in Bude where I will stay for 2 nights. After that my friend will drive me back to Exeter. I will take another train to Manchester and catch my departing flight back to the USA. My flight departs at 10:40am so I will be staying overnight in Manchester. I will book these train trips very soon.

I may have packed too many places and too much transit into my 2 week holiday. Too late to change plans since my transatlantic flights are booked.

If you're interested in how I spent two nights in Manchester, as part of a trip to Glasgow, Manchester, and Liverpool, here's my trip report, including details of my accommodations in Manchester: https://community.ricksteves.com/travel-forum/trip-reports/uk-trip-report-glasgow-manchester-liverpool-in-september-2016

Thanks a lot Harold!