3 people--12 days "on the ground" in Italy

I would suggest checking lodging at booking.com to find places with the amenities you want, there are plenty in your price range.

You could go to Cinque Terre after Venice. Good idea to fly open jaw.

Venice Hotel - if you don't mind staircase - Hotel Tivoli is a modestly priced hotel with a triple room. The room has a small private bathroom, decent breakfast, is clean. It is a no frills place. It has AC at night and a TV. Good location. I stayed in the triple room last June. Rome - you may need to increase your room budget to get a good location. Check Hotel Smearaldo. Good location - 5 minute walk to Pantheon. No frills place. I stayed in a different hotel close to Smearaldo. Try Hotel Pendini in Florence. I stayed at Pendini and got a very good price because we stayed 5 nights. Compare hotels.com and booking.com.

I'd say go to Florence first. The fastest way to Cinque Terra from Venice is though Florence. 5 1/2 hours with changes at Florence and Pisa. So you might as well break the trip at Florence. There are a few Freccia express trains direct to Rome from La Spezia (jumping off place for the CT) bypassing Florence.

There is a second best way taking the train back to Milan and then from there to Cinque Terra. It is 6 1/4 hour but only one change. So that would be the way to go if you wanted to do the CT first.

I would do CT in between Florence and Rome.

The best train connections take you from Venice to Florence to the CT to Rome. Anything else will be backtracking somewhat. My guess in that there won't be much, if any, difference in crowds on weekends.

It may be a little early to find hotels for June. Many smaller places don't take reservations that far in advance and may not even have their 2015 rates yet. So if you find that sites like booking.com and venere.com (good for Europe) are showing "no availability" check back every couple of weeks - or contact the hotels directly.

You're making progress, as you now have your flight booked, but the rest of this question is pretty similar to the last three questions/threads posted. Chani's description of the order of train connections makes sense.

" This way we wouldn't be in CT on a weekend (I'm thinking it won't be as crowded during the week)"

During your time listed, any day in the CT will be crowded, but GO anyway. Here is a good place to stay in Vernazza:

http://www.elisabettacarro.it/

Have a great trip!

We enjoyed this apartment in Manarola for its space and location. Nothing fancy but good WIFI, big terrace with sea view, nice host.