I've only ever passed through Genoa and never stayed but I have heard so-so things about it as a tourist city. I'm sure some people will jump to defend it but it is not generally a city tourists seem drawn to stay in so I might advise you to look into it as a city before just deciding to stay there.
Either Milan to CT or Lucca is a long day on the train - I've done Varenna to La Spezia in a day and it will be kind of a trek. Keeping that in mind I would choose CT next because otherwise you're doing inland to Lucca, back to the coast and then crossing back inland to Verona etc.
If you are really interested in the lakes I would advocate not necessarily use Verona as a base for Lake Garda. The lower part of the lake is easily accessible by train but that means it is touristy as possible with water parks and the likes with the original interesting towns kind of buried underneath all that. If you want to really explore the lake I would advocate for taking the bus and visiting towns up the east side of the lake. Malcesine has access to the upper lake ferry and a cable car up Mount Baldo.
If I was putting this together I would think about Milan > CT > Lucca > Bologna > Lake Garda > Verona > Dolomites > Lake Como > Milan (fly home) but you also don't mention how many nights you have in total. I would say this is a minimum of 21 nights and probably a little more to give everything time. If you have less I would probably drop one of the lakes or Lucca and work from there.
Lucca is your only Tuscan destination and I really like the city, but not everyone likes it as much. I would advocate at least two nights there (one full day) or better three (two fulls) to explore the city. It's easy to include Pisa if you want big sights part of your visit because Lucca has quieter appeal.
Verona is an upscale, mostly pedestrian, chic, arty city with an intact Roman Arena and theater tucked in the bed of the river. It has a "cafe culture" feeling to it and feels IMHO much smaller than its population.
Bologna is a big city with an ancient university, and old city core and is the food capital of Italy. It has a younger and hipper vibe than many Italian cities.
I'll leave the Dolomites to someone else but I know many people plan on renting and returning a car in Verona as part of this plan.
Hope that helps, have fun figuring it all out,
=Tod