I've only passed through, but there is a children's amusement park called (not very originally) Gardaland which should appeal. Note that many Germans visit Lake Garda, but the Brits are welcome too.

You've touched on the BIG problem with the area - access. The convenient airports are Bergamo (which Jet2 and Ryanair call Milan) and Verona but there isn't much to those. On top of that is the increasing problem of Tour Ops cutting back on their range and Italy being dropped. Italy has become quite expensive so isn't as useful in the bulk market.

From the big names TUI still have 2 specialist departments which may be of interest, see http://www.citalia.com/destinations/Italy/Italian%20Lakes/Lake%20Garda and http://www.crystallakes.co.uk/destinations/lake-garda-resorts/index.shtml

The lake shipping management have provided a useful English guide to the lake at http://www.navigazionelaghi.it/eng/gar/nlg.html and you can work your way through the list of towns on the right.

If you went DIY you'd need to add in a car with kids that young and then it would get expensive. A couple of adults could reach a good range of resorts by public transport but it's a bit hit and miss. Your kids are of an age where the location would be different enough to be interesting (I take it you don't have many mountains at home!) so only a few added facilities would be needed to keep them happy but once they reach secondary school ages it will become less appealing.

I think as first timers the best option would be to find a resort with a few facilties and then find a half board package tour.

Hi
I think Peschiera del Garda would suit you
Featured in Thomson lakes and mountains and offering properties with mini clubs for children aged 6 to 12.
Charter flights into Verona from Birmingham on wednesday and saturday.
Inghams also feature Peschiera with 2 suitable properties for your kids.
Flights with Ryanair into Bergamo from Leeds Bradford on wed and sat.
Peschiera is close to Gardaland and Caneva World theme parks.