I have lived in several parts of the world, and travelled to many more. Particularly countries in Europe and Asia. I was born in, and currently live in Australia. I have visited Italy a number of times.
My current main camera is a Canon 7D, and I have a few 'backup' (older) XXXD (Rebel) series. My main lens is the Canon 15-85mm, which provides a wonderfully flexible focal length range. I have many many lenses, including the 10-18mm and 24mm STM. Both these are favourites for travel (light, and IS in the UWA).
For Italy, I think having an UWA is definitely high on the list. For a 2 lens setup with my family, I would personally take 10-18mm STM and then also the Canon 15-85mm as the all purpose, and for candids (and 'family shots'). If it was to grow into 3 lens setup, I would probably add a telezoom (I take either a 55-250mm STM or my larger, but still portable 70-300mm L). The 70-300mm L is a high quality lens, though the 55-250mm STM is no slouch either. I agree with earlier comments that travelling light and enjoying the time away (with wife and children, and/or friends) - is very important.
Regarding the 24mm STM, I love this lens. It provides a great FoV on an APS-C / crop sensor DSLR. Bbut I am not sure that I would take it on holidays, as I have that focal range covered with my 15-85mm, and in some sense it is possible you could 'cover it' with the 18mm tele end of your UWA (10-18mm STM), and go for something more tele (e.g. 85mm range).
I regularly organise photography outings (including weekend and all day outings) here in Australia for friends (and 'informal students'). For these occasions I pack a LowePro backpack full of glass and accessories, and usually take a tripod along. But when I am travelling on holiday with my family, I avoid taking a tripod if at all possible. While 'tripodless' (obviously) limits night shooting, I appreciate IS for some low light / possible shots. Or use walls, seats, other solid objects for the occasional long exposure.
Taking 2 (even identical) cameras (if you have them) can be important, so you don't need to use (or 'loose') your and your family's (!) time when you change lenses... and often 'change the lens back'. Having one camera in your hand/ round your neck, and the other in a small shoulder bag 'ready for grab and shoot' works (very) well for me.
Enjoy your holiday, Cory, and time balanced between quality time with your wife and 2 children, as well as the 'photographer' experience!
Paul 8)