I would look at SLC-ATL-FCO on the outbound, then FCO-AMS-SLC on the return. The SLC-AMS routes are flown by DL on a 767 or KLM on 787. For the DL flight, the timing of the outbound makes it a bit odd for sleeping during the flight, and likely gets you to Rome too early to check-in to your a hotel. The KLM flight is at a much better time, but the 787 in economy isn't very comfortable(not as bad if your sharing your row with children/smaller family members). Theres numerous SLC-ATL flights every day and the ATL-FCO flight is a better flight in Y on the A330. For the return, similar story, but flying straight home from Europe as opposed to connecting back in the US is always nice. That being said, in both directions, I would prioritize overall flight times and connection times over any specific routing. With little international flying experience, I wouldn't book anything with less than a 90 minute connection when dealing with passport controls. Longer connections just mean you can get a decent meal on the ground vs what you'll get on board.

Also, just doing a quick search on the website, I'm seeing one-stop options with SLC-DTW-FCO, and SLC-ORD-FCO with ORD-FCO operated by Alitalia. Not sure what routes are seasonal though so depending on OPs dates, these might not all show up.