You could easily spend 3 weeks just in Italy.
We like a mix of a tour and time on our own. This gives us some downtime from being "on" with a group, as well as time to see some things not included in the tour. Our experience with RS and other tours is that they are very active - you are up and out the door at 8 or 9 am, every day. Some days are all day touring, some days have longer bus travel, some days have ample free time. Check the itineraries of the ones you are interested in. Personally, I think back-to-back tours, unless they were short, would be a little too much go-go-go for me. YMMV.
So we tend to book a tour, and then add days before and after, although we try to make it fairly simple. For example, we did a Rick Steves Portugal tour and just added days in Lisbon before the start, and then added days in Porto, which is where the tour ended. The tour was 11 days, we were gone a total of 18 days. There was a lot in both cities that was not included in the tour, so that worked out great. We did an OAT tour of Greece and Croatia, and added days in Athens prior to the start of the tour (the tour - a small ship cruise - did not include much of Athens, and we wanted to see the major sites). That tour was 16 days, and we were gone for 21 days.
My advice is to decide if you are booking a tour, and then plan from there. Some are 7 days, some are 16 days or more, so that's a consideration. And make sure you factor in travel time with plenty of cushion for late flights, cancelled local trains, etc. Stuff happens!