Italy bakes in the summer so if you have 2 smaller children, early in the summer (May/June) or late Summer would be best - I'm assuming that neither are school age? Italy is our favourite place and our 3 DD's have been dragged all over. Italian cities see a lot of tourists in the summertime which can make moving about them slow and annoying, particularly when it's really hot - Venice for example even in springtime with a buggy was really really difficult! Bologna with a buggy was much better - less crowded, less expensive, less big and with portico's covering the pavements so even when it was hot, there was shade.
An easy first trip would be to fly into Verona (a number of the cheaper airlines go there, it's a 2 hour flight from London), spend a couple of days in Verona (you may find an apartment easier if either are fussy eaters - try AirBnB) then take the bus from the railway station (takes about an hour and they go very regularly) to one of the towns on Lake Garda. There are all sorts of accomadation types around the Lake from campsites through to hotels so you'd find something to suit you! You can also take trains from Verona (train travel in Italy is cheap, safe and frequent) - Venice is reachable in just over an hour. Verona/Lake Garda are further North so while they do get hot, it doesn't get quite as hot as the South.
In terms of other places that might suit with kids of your ages, the Amalfi coast is beautiful with gorgeous beaches - it is pretty expensive though but if you have the cash then you will find some great places. A package deal with transport included would probably be the easiest with 2 small children but there are public transport options from Naples if you want to save cash and are confident travellers.
Another nice trip we did was flying into Nice (you can get some very cheap flights, again only 2 hours from London) staying a couple of days there and swimming off the gentle beach in Villefranche (a 10 minute train journey) and then taking the train to Sanremo from Nice (it's about an hour 20 journey). Sanremo is on the Italian Riviera - used to be on the super rich yatching route and is very much a place of faded glamour, beaches and palm trees. There's a nice old town (parts of which might be too hilly) there too as well as casino's and the like. From Sanremo you can catch a train to Genoa and fly back from there.
One place I would totally avoid is Lido di Jesolo off Venice. I took my eldest DD and nephew when they were 4 and it was the worse holiday I've ever had! Lido di Jesolo is a sand split built by Mussolini and is basically is a long stretch of beach with lots of 40's/50's hotels behind it. There was a number of run down parks, all the hotels looked like they'd been thrown up with no thought and it was pretty souless and ugly (IMO!). Venice is a bus and then ferry ride away which was a pain. It looked ideal for 2 small kids but I hated it - to be fair though, both my DD and nephew behaved incredibly badly and we had no AC which probably played a big part in it!
Sicily is lovely but of the places we've been, they've not been that buggy friendly. I've not been to Sardinia - I've a friend who goes there frequently as there a number of AI places and it's less hot in August. I don't think she's seen a great deal of Sardinia outside of the hotels at all but horses for courses!