After reading all the posts, I agree that we prob do not have too much
time to do a day trip and will focus on sights in Rome as we will be
there 3 nights as well.
That's a good decision as 3 nights is, at most, 2.5 days or even less if not arriving until the afternoon. Where will you be coming from, and what does your FULL trip itinerary look like? As Tim said, no evil hackers will be able to track you down if you share that info, and the bigger picture can help us make the best suggestions for you plus look for any pitfalls in the plan. :O)
The Colosseum is a toughie: it is THE most-visited attraction in Italy, and tickets for high and shoulder seasons have become increasingly hard to land over the past number of years: you're going to be fighting with thousands of other folks for them.
These recent threads include some info about ticketing, and illustrate how quickly they sell out once they become available on the official vendor's site (https://www.coopculture.it/en/index.html).
https://community.ricksteves.com/travel-forum/italy/colosseum-ticket
https://community.ricksteves.com/travel-forum/italy/forum-palentine-colosseum-tickets
https://community.ricksteves.com/travel-forum/italy/rome-colosseum-ticketing
There are also independent companies that offer tours. A few of those which have been positively recommended on the forums are:
The Roman Guy:
https://theromanguy.com/tours/italy
Walks of Italy:
https://www.walksofitaly.com/?gclid=EAIaIQobChMIlZaok82p_gIVkUVyCh0K-QroEAAYASAAEgIAo_D_BwE
Through Eternity:
https://www.througheternity.com
I'll caution against booking tours, outside of those offered by coopculture, which include the underground as there's no guarantee that even the independent companies can get hands on those. There is a HUGE demand for them, and you'll fare better booking a tour which does not include it as there's the possibility it could be cancelled if they can't get tickets. I'll suggested booking with one of the independent companies I listed for a 3-4 hour tour which includes the Palatine and Forum - which are right next door to the arena - as well as the Colosseum as those two archeological sites are more complex and benefit from an educated human guide.
Regarding Coopculture: You cannot buy tickets to the Colosseum, Forum or Palatine AT the sites - only online - and all ticketing for the Colosseum is timed-entry, meaning you have to choose a day and time slot which still has tickets or tours available. With 6 people in your group, that's going to be really difficult on the Coopculture website so explore choosing a tour from one of the independents.