Transportation from train station to the hotel

Where are your hotels? Many are walking distance of the main train stations.

Depends where your hotel is. In all 3 cities the railway station is on the edge of the historical centre. If you pick the right hotel, it would be within walking distance of the station.

Venice - this is a special case, there are of course no roads. The options are walk, a Vaporetto (water bus) or a water taxi (expensive).
You will need to find out about Vaporetto tickets for your stay.
A good option is a hotel near the station. Walking cases through Venice is not easy as you keep having to cross canals on bridges which have steps up and down. St Marks square is at the opposite end of the islands to the station.

Florence and Rome Walk or taxi. Rome has buses and a small metro, but these will not be optimal when you have luggage.

My first move is to contact the hotel. Often times the hotel's web site will give directions for getting from the train station, airport, or highway to the hotel. The fall back is always taxis if walking is out of the question.

Go to rome2rio.com and enter the train station and your hotel address. The column on the left will show you options -walking, bus number, taxi, etc. The map gives a visual of the route.

I have used Rome2Rio in Venice and ignored the suggestions as they do not work well. I don’t hink Rome2Rio is able to wrap its head around vaporettos.

One app that works perfectly in Venice is CheBateo (which boat) which links to vaporetto timeables. That app is what Venetian residents use.

I have this same question (pertaining to Florence). It's just under a mile to get from the train station to Piazza della Signoria. I have no problems walking a mile, but with luggage - though not huge and easily manageable - do folks regularly walk this distance? Is it the norm to walk from the train station to a hotel in Florence?

I live in a hilly city, so perhaps Florence is flat! ;)

Well, depends on how you're feeling that day, the time you have, the weather, etc. Taxis are always an option, and your hotel can make the arrangements ahead of time. Buses or trams might be availablem if you're comfortable with them. Avoid? Note taxis in Europe dont roam around waiting to be waived down, as in the US. They wait in designated areas, or come when you call them. Dont get in an unmarked, unmetered cab, which is good advice in the US as well.

I feel like these questions also need to include two vital pieces of info: your mobility level, and as importantly, carry on luggage vs steamer trunk? The recommendations would be very different in each circumstance.

In Venice in my dream life I'd be like George and Amal, cruising down the Grand Canal with my custom monogramed hat box and dropped at the door of my hotel. In my real life I have a tiny carry on bag, and just want to get where I'm going in a cost effective manner. In Venice there is another consideration - is your hotel anywhere near a canal that allows for water transport? Little back canals aren't accessable by water taxi and may not be near the main Vaporetto routes. Getting there by gondola might be possible but will be costly - I'm not sure they'd even agree to carry luggage. I've never seen that). If your hotel is several back alleys away from the main water routes it doesn't matter what method you chose, you'll ultimately be dragging your suitcases (another argument for carryon bags only, but I digress...)
In Venice we've used the Vaporetto, but our family of 4 only travels with carryon. We were each time in peak tourist season so the boat was full. Very glad we only had a small bag each. The boats circle the island in both directions so we took the route that was the longest to get to our hotel near Piazza San Marco. We wanted to get a free sightseeing tour on the way to the hotel. From the stop where we disembarked it was at least a 5 minute walk with a couple of bridges (read stairs) to cross on the way. There are multi day passes available so figure out if you'll want to use the vaporettos other than getting to and from the train. A multi day pass is handy if you want to see all parts of the island and reduce your walking - jump on a boat for a few stops and get off to see a neighbourhood, then jump on another to go to a different part of the island. Next time I'm in Venice I plan to stay closer to the train station, not because of the distance issue but to be based in a different part of the city than last time so I don't need to rely on the boats.

Been to Rome 4 times and stayed in two different hotels on Via Nazionale. Within 15 minutes walk from the station. Used a taxi once when we were cutting it close on catching our train out, but normally we'd just walk with our carry on bags.

Haven't been to Florence yet (planning for fall 2019). I'll be watching for others' recommendations.

I normally try to pick hotels that are reasonably close to the rail station, but there's usually some walking required. When I'm hauling luggage, I usually use a Taxi to and from the hotel.