As a Denver-area based traveler, I'm not familiar with what airlines serve the (San Francisco?) Bay Area, but are there any actual direct flights between your home airport and Rome or Venice? If so, that would be the way to go, unless the cost or flight times were somehow prohibitive.
Otherwise, you're looking at a mid-flight connection somewhere, either in the USA or in Europe. For Denver International Airport, our airline options for Europe for a long time had pretty much been Lufthansa (connecting thru Frankfurt or Munich), British Airways (connecting thru London), or United (connecting thru Dallas, Chicago, etc. -a pain and not a comfortable as either the German or British planes). In addition to those, now, Denver has Icelandair (with a usually short connection in Reykjavik) and Norwegian Air (with a connection in either London or Paris).
Since cost is always a big consideration for us, one must be cognizant that some of the newer airline options can come with extra fees for luggage and on-board food and beverages, and baggage weight restrictions. Any of the above airlines will get you there in reasonable comfort, but check the connection wait times and overall flight durations. Also, if you have to transfer from one airport to another (London Heathrow to London Gatwick, for example), that could influence the overall travel time and cost/convenience.
If you use one airline to get to an initial, general European gateway, and then another to actually get to Venice and/or from Rome, that's another possibility, but you probably won't be able to check your bags all the way through, and you'd need to claim them, go through customs, and then re-check them. But budget European airlines like EasyJet and Vueling (and others) offer quick, convenient, cheap flights within Europe, and we've used them to sometimes do a multi-airline method of getting to/from Europe.