My wife and I are finally going to Italy this year. The last two weeks of September. Flying round trip to Rome with the intention of heading north to Venice on 1st day of arrival and then working our way south. Areas on our list are Venice, Florence, Cinque Terre, some of Tuscany & Umbria, down to the Abruzzo area (wife has distant relatives in Teramo), down to Naples, Sorrento and then back to Rome to finish.

I am looking for recommendations on booking accommodations. Budget goal is to try and stay under 200 Euro a night (although I know that may be tough in areas). What methods/people have others used successfully?

I can't recall the names of each lodging, but I wrote 2 different trip reports which included Florence, Cinque Terre, Sirmione, Verona, Chianti region etc. I do remember Hotel Baia in Monterosso (Cinque Terre) was a great spot, we had a 2 room unit with balcony for about $200? We used Airbnb and Bookit.com primarily. Have fun planning!

I cannot help you with all your locations but can throw out a couple places.
However, I will say you may need a vacation when you return from your vacation… you guys will be covering some territory.

In Florence we stayed at the https://www.sangallopalace.com/
Very nice hotel and reasonable. It is not in the thick of things. About 12 blocks to the center of town, but far enough away to escape the mass of tourist that over take the city. Great neighborhood restaurants around the corner. There is also parking nearby, which is a consideration in Florence if you are driving.

For the Cinque, which is an amazing place, if budget is a consideration you may want to consider La Sepzia. We stayed at My One hotel, but now under new management and is called the CDH Hotel La Spezia. Great Location! And once again under your budget. You are two blocks from the ferry to the Cinque. However, I would not skip Lerici or Portovenere when you are in that area. All three places are a very short ferry ride from La Spezia. La Spezia is a real town and not very touristy, so at night you will get to experience the Paseo. In the evening the children come out and play in the square, grandparents gather and watch them while the parents sit in the cafes enjoying adult time. Then the family will walk together through the streets and alleys for dinner which is served promptly at 9 pm. It is not as romantic as the Cinque... but not sure anything compares.

You are going to have sooooooooooooo much fun!

I will second Beerman's recommendation of Hotel Baia in Monterosso - we stayed there this past September & loved it. We did Airbnb in Rome at 2 different places & Florence -all were fantastic. If you are interested in a tour of the wineries in Tuscany, I can recommend Tuscany Wine Tours with Angie, https://www.angelapersonaltuscantour.com/ - our original plan was to rent a car & do a wine tour on our own, but had second thoughts about drinking & driving. We flew from Austin-Dallas-Rome & then took trains to Florence, Cinque Terra & back to Rome (hence 2 different Airbnb's in Rome to catch our flight to Poland. We were a little hesitant to take the trains, as we knew no Italian, but had absolutely no problems. Made all our train reservations before we ever left home. We also bought our tickets for the Colosseum, Roman Forum & the David statue before leaving home. Please feel free to message us if you would like to ask any specific questions. The Rick Steves travel books are a wonderful help - we use those to travel quite a bit throughout Europe.

Paris Hotel in Florence, Italy is going to be a good option. I know a couple who stayed over there and I just checked the tripadvisor reviews for you and it has got good rating too. Also found the recommendation for Hotel Perseo while browsing the forum.

We did 10 days in Italy Sept to Oct 2018, expecting smaller crowds. Rome and Venice were jammed packed.
We only did 1 night in Rome staying in an AirBnB within walking distance of the Forum @ $85CDN per night. We should have stayed longer and will return for at least 3 days. We lined up first thing in AM to tour the forum $80 Euros each) and at 1130 we were still in line when they announced that it was closed due to being maxed out. It is the 2nd most visited tourist attraction in the world. Great Wall is #1. We had a train to catch to La Spezia and were fortunate enough to find the guide who sold us the tickets and got a refund. If you are in the Forum area at sunset make sure you go to the rooftop bar at the Hotel Forum.
In Riomaggiore we stayed in an AirbBNB for $75 CDN, it was the only disappointment of the trip. Huge 2 bedroom apartment but the bathroom had no traps so smelled like sewar gas. I blocked all drains with baggies filled with water to no avail. But there are tons of AirBNB's so consider them as option. Day 1 we hiked from Monterosso and that just about killed me...it is a difficult trail. The other easier trails were closed for repairs. Day 2 we went back to Monterosso and spent the day on the beach. Day 3 we caught early trains to La Spezia and then Pisa for breakfast and then on to Florence. Pisa tower was about a 30 minute walk from train station. Plan on buying your leathers in Florence. We then headed by train to Venice where we had the best AirBNB ever...view of the canal right by the Venice Hospital. Contact directly, avoid AirBNB fees, Lotus Apartments Venice – Ca’ Laguna, e-mail lotusapartmentsvenice@outlook.it Venice was a lot cooler then Cinque Terre and the other places. Shorts yes but needed a shell / sweater.

Thanks to everyone that replied. Some good information to get started with.