Well, it is possible to squeeze in 1 more destination, however it all depends on your sightseeing practices & interests. If you are not that much into museums and places far from the city centers, then you can more or less manage to see majority of attractions in two days per city. Obviously one can spend much more time in each of those, but for the first visit it might be ok itinerary (at least it will definitely give a good feel of what you like in Europe & to what kind of places you would prefer to come back
Rome to Florence: easy & cheap. If you ok buying discounted (non-returnable) tickets in advance, then you can get good deals for as low as 9 euro per person. There are two big railway companies serving the route - state owned
Trenitalia & private operator -
Italo, so you can check which one is cheaper for your preferred time.
Florence to Venice: more or less the same story as above, buying discounted ticket in advance can get you a very good price
Venice to Switzerland: here it kind of depends on where in Switzerland you want to go (mountains? Zurich? Geneva?), but in general you can start with the same TrenItalia website to see rough travel times & connections available (majority will be with change in Milan)
Switzerland to Amsterdam: it won't be very fast by train (at least 8-9 hours), so perhaps it might make sense to look into flights here (--> plenty of lowcost options, e.g. EasyJet directly to Amsterdam from Basel or Geneva for as low as 20-30 euro)
Amsterdam to Paris: probably won't be super cheap, but still relatively easy, since you have plenty of direct high-speed
Thalys trainsParis to Manchester: yet again, just like with Switzerland to Amsterdam, it is possible to travel the whole way by train with change in London (
EuroStar Paris-London & then
domestic train to Manchester), however it will take you >5 hours. So flying might be a good alternative option here as well