Visiting Cinque Terre, Venice, Florence, Tuscany, Rome, Soreento(Almalfi Coast) in 3 weeks

This is doable in 3 weeks. Do you plan to rent a car or use public transportation? Be aware that September is a very popular tourism month for Italy so accommodations will book up fast.

The RS Italy guidebook, or any other reputable guidebook, is a worthwhile investment compared to the dollars you're spending to make the trip. It will give you the essentials of how to get from one place to another and it will list the more popular sights to see in each destination. It will also list hotels and B&Bs with prices, so that even if you end up booking somewhere else you will have an idea of what's a reasonable price for a given type of accommodation. (I'm not affiliated with the RS organization or another travel company.)

If you are able to fly open jaw -- arrive Venice, depart Rome -- you will avoid backtracking. The reason for not doing it the other way around (from what I've read) is that Venice flights to the US tend to depart early in the morning and it is difficult to get to the airport from Venice proper (as opposed to the mainland) at that hour. The other reason is that over the course of 3 September weeks the autumn weather will move in, and if you're moving south you will likely enjoy summery temperatures on more days of your stay.

I did a trip very similar to this years ago as my first trip to Italy. Train is the easiest way to get from place to place. You might consider renting a car for Tuscany (outside of Florence) for a few days or for Amalfi, but it's not required. (Actually, I've never been to Amalfi though I did spend a few nights in Sorrento.)

I'd fly into Venice, out of Rome or Naples (whichever makes sense - Rome is probably cheaper and easier, especially if you can get a direct flight from SFO) as an open-jaw "multi city" ticket, not two one way tickets. If flying into Rome to start is a whole lot cheaper, you could consider taking the train immediately to Venice, which is what I did the first time I went to Italy. I would try to end in Rome though and fly out of there, not split your time in Rome.

My itinerary was: Venice, Bologna, Florence, Cinque Terre, Sorrento, and Rome. The train trip from Cinque Terre to Sorrento was the longest with a few connections, and there was really only one train itinerary that day that worked. Just check the schedules and know what you are dealing with.

To save money staying in the Cinque Terre, consider staying in Levanto, the town just to the north of Monterosso, the "fifth" village. It's only one train stop away, and though I wouldn't expect cheap lodging in Levanto in September (still a busy time in the CT), it should be cheaper than in the other five villages. Try Booking.com and AirBnB and see what you can find. I'd book things ASAP as a first pass that can be canceled for free til near the last few days, then as you get time go back and check lodgings again and see if you can find something better.

Certainly doable. Start by searching airfares--into Venice and out of Naples if you can find it.
In the meantime, work on defining what you mean by "Tuscany," as it is a large area with many many options. Determining if you will rent a car for that portion will help cement your plans, as it's all easy (though time-consuming for that CT bit) by train. The notable exception being that once you get to the Amalfi Coast, you switch to ferry or bus. Also work on exploring if you wish to stay in Sorrento or one of the Amalfi Coast villages.
Though there are differences, I find visiting both the Cinque Terre and Amalfi Coast redundant, so you may want to put more time in Tuscany or some of the cities to avoid that.
Once that research is done, all will fall into place.

I think the biggest challenge is between Venice and Cinque Terre what
is the best way to travel and then on to Florence.

No challenge at all: the best way is by train, as it probably will be for all of your other locations. For this many destinations in 3 weeks, and not knowing the number of NIGHTS you'll have on the ground in Italy, I might do something like this (in order, using 20 nights and not including your overnight flight from the US):

Fly into Venice
Nights 1 - 3, Venice. 3 nights/2.5 days
Night 4 - 8, Florence. 5 nights/4.5 days with some day trips vis public transport (Lucca, Siena, 1/2 days to Pisa and Fiesole, etc)
Nights 9-11, Cinque Terre, 3 nights/2.5 days
Nights 12 - 16 Sorrento OR Salerno, 5 nights/4.5 days with day trips to Pompeii, Capri, Positano, Amalfi, Ravello
Nights 17-20, Rome 4 nights/3.5 days
Fly home from Rome.

Florence is in Tuscany, and basing there cuts the number of hotel moves down. Siena is best done by bus as is Fiesole. You could shift a day from Florence, Sorrento/Salerno and/or the CT to Venice or Rome, if those two cities interest you more but this would be the trip I'd probably do. You could also use those shifted days for a short stay in one more location as well, if up for another move.

The CT is expensive so if you find accommodations are pricing beyond what you want to handle, look at Levanto. I would book something SOON as hotels in that area fill far in advance. I would say the same for Sorrento so I gave you the option of Salerno as it will probably be more budget-friendly.

Thank you everyone for your advice. I have just learned of these Jaw tickets, so I need to look into that. I looked at booking the Trip through Costco as it is over whelming me. Normally I go on vacation I stay one or two places. Even looking at Rick’s web-site and book to look at hotels in Venice it asks what area, I have no idea I have never been there. I imagine it will be the same for each city. I like to be by water, have a view etc. where I can. Any recommendations on hotels would be greatly appreciated, I don't know if it's possible but would like to stay under 100 and have a great place :)

I saw one beautiful hotel in Sorrento, Grand Hotel Royal
and in Venice, Papadopoli Venezia

NH Collection Venezia Palazzo Barocci

Starhotels Splendid

have not had a chance to look up prices but the seemed beautiful, again area I have no idea if they are in good area's. I like to be close to the action but not right in the middle of it. Thanks

I saw one beautiful hotel in Sorrento, Grand Hotel Royal

Yes, it's very lovely and also a 5-star hotel with rates to match! Just a random price check on their website of 5 nights (Sept 18 - 22, checkout of the 23rd) the lowest rate was $2468.84, no sea view. If that's a bit rich for your wallet, you might want to get onto booking,com and search by the per-night rate you're wanting to spend? Also make sure that your accommodations are close to city center and not WAY out on the fringes: you can check locations on maps on the menu of each hotel's page.

$100 U.S. dollars at current exchange rate is 81,07 euros; very low for a hotel room during high season, especially for anything with a "view" so keep the exchange rate in mind when looking at prices? Also ignore the random prices in the headlines of TripAdvisor - such as "Hotel Papadopoli Venezia MGallery by Sofitel $191". They are advertising current OFFSEASON rates for select nights. 3 nights Sept 7-9 comes up as $1,213 on booking.com so best to adjust your expectations. :O)

Yes, looking at a website like booking.com for your specific dates will provide a reality check about what is available in your price range. Italy is not a dirt-cheap destination to begin with, and all your target destinations are relatively popular ones. Plus, the dollar is considerably weaker than it was last year.