Archaeology interests my husband and son so I wondered about the
church Basilica of San Clemente. Have you been there?
Yes I've done it and I'd definitely include that one if one has an interest in Roman history/archeology. The current church - well worth seeing itself - is just the tip of that iceberg! There is a fee for touring the subterranean levels.
http://www.basilicasanclemente.com/eng/index.php/informations/excavations
I would love to climb Palatine hill if we had time. Have you done
this?
I'm not exactly sure what you're asking here? The Palatine - which yes, I have done - is part of the Colosseum/Forum/Palatine complex and involves a ticket to see. Most tickets to the Colosseum include entry to the Forum/Palatine as well but in what order depends on what sort of ticket or tour you have.
This is a rather complicated subject and time can be of the essence for buying Colosseum tours/tickets as certain of them sell out very quickly. Budget also comes into play as well: tour prices vary depending on just HOW much of the Colosseum you want to see, and if you want a guide for all three archeological sites. I might recommend the latter option if one has a keen interest in archeology/history as the Forum and Palatine, especially, are complex sites that benefit from a knowledgeable guide.
ONE CAUTION: as you are researching your options for the Colosseum/Palatine/Forum, do not use a guidebook for gathering your ticketing/tour info, and be careful of just how old online info - even on these forums - can be. There have been many, many changes to tours/ticketing options for these three sites this past year so there's a LOT of outdated info out there. There could be even more changes coming for 2020. This is the official website and best source for most CURRENT info:
https://www.coopculture.it/en/heritages.cfm?id=3&id_c=58091
Tours provided by independent companies - such as Walks Of Italy, The Roman Guy, Context, Dark Rome, etc. are also available.
The colosseum and Roman forum ( maybe Capitoline museums as I believe
near here) is a definite and the Spanish steps and fountain
The Spanish Steps and Trevi fountain are merely walk-bys: do those on arrival day as trying to do museums or major archeological sites in a jet-lagged state is not advised. Suggestion? Many guidebooks, including Rick's, provide self-guided walking tours. These are good fresh-air activities for staying awake and also for getting your bearings in the Eternal City.
Whenever you might choose to do them, your other children may not find the Capitoline Museums all that interesting. Some of the many churches posters found interesting are listed in this thread:
https://community.ricksteves.com/travel-forum/italy/churches-in-rome