From personal experience, as long as Rome Fiumicino (even pre-fire) airport is operational, every other Italian airport will be at worst a distant second in the incompetency sweepstakes. Napoli Capodochino airport, even on a Saturday morning, was a pleasure in comparison. Fiumicino can be beyond awful.
Some misinformation upthread. First off, our Fresciarossa train from Florence SMN to Salerno station left at 12:04 and arrived in Salerno at 15:42, after stops at Roma Tiburtina, Roma Termini & Napoli Centrale--no train change, either. That's pretty efficient to my way of thinking--the train flies at speeds up to 270 km/hr, and smooth as silk. Weather-wise, after checking monthly averages (which really don't mean squat in the grand scheme of things) temperatures are 2 degrees warmer in Amalfi than Florence, and Amalfi averages 9 days of rain in April as opposed to Florence's 8. So pick your poison, as it were. It might rain a bit, in either area. Buy an umbrella.
So...it's really an issue of not running around like a chicken with your head cut off for eight days by lodging in too many places overall, three at the most. That's why daytrips are key for you, and why Florence is a great base, as well as Salerno or Amalfi. When we were in both places in March of this year, for 5 nights each, we had Siena, Pisa, Bologna & Fiesole planned for daytrips out of Florence, and did none of them, opting to groove at our VRBO apartment in town--what fun, not having anything we had to do! In Salerno, we stayed at a great, inexpensive B&B, planned daytrips to Naples, Paestum, Amalfi & Pompei--and did the last two, less than an hour each way in travel. And we'd return by 5:00 each night to Salerno, take a nap then a walk down the promenade by the sea, and go to dinner in Salerno's Old Town when the restaurants opened at 8:00. Quite nice, actually.
You cannot go wrong with this trip, but remember, variety is great & less is more sometimes. Enjoy!