Seems like a sensible overall plan.
The challenge would appear to be the transfer from Firenze to somewhere on/near Lake Garda. It would be a costly taxi journey. The train does look the best option, with a direct train from Firenze to Bergamo that stops at Peschiera del Garda Sirmione. If you'd be ok with that train journey, then taking a taxi from there to your destination would be much cheaper (and nicer than a long car journey).
Booking in advance on Trenitalia can be very cost effective and trading up to 1st class still means it will be rather cheap. We've not asked for assistance at train stations, but I'm sure that will be available, and definitely worth asking for. 2 hours is a lovely length of journey on these fast trains, and for us it remains a joy rather than a hassle. The 11.36am might be the best option, arriving at Peschiera del Garda Sirmione at 1:34pm, which by the time you've got a taxi to the end destination, should make it a convenient time to check into your new base.
If worried about the train, I'd be happy to talk you through what to expect, and some little tricks to help e.g. identifying the usual platform in advance, and even the layout of the train, so you can stand roughly near where your reserved seats will be. There are probably people here who have even caught a train on that line before.
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Lake Garda does sound like a good choice, with boat trips and local buses for transport. If however the Dolomites call, I'll put in a good word for Trento, which is reasonably flat (at least in the centre), has a lovely spacious feel, and plenty of charm (some friends described it like Verona before the tourist hordes took over). There is a cable car that goes up above the city, a lovely old thing it is too. Buses can get you up to the mountain villages / resorts e.g. Fai della Paganella, Molveno, etc., and there's always the train down to Rovereto. Molveno is stunning, with lake, picturesque town, mountains and no 'filler'. It is breathtakingly scenic.
In terms of transport , there's a direct (4 stop) train from Firenze to Trento at 12:36, getting in at 3:15pm (with Rovereto the stop before if that appeals). Those trains are lovely, and very much a joy to use. There are taxis at Trento station and any taxi journey will likely be short and cheap.
Bologna or Verona worth considering as exit airports if they're an option for you, as both are well served by that same fast train from Trento/Rovereto. Verona also an option if you choose Lake Garda.
