Should I get these online in advance? If so, what website?
Yes. look at the excellent youtube channel ROMEWISE for good information on the various tickets, etc.
Yes, absolutely you should buy in advance, and you should do that via the official website:
That part of the park's website - largely informational - is only 'live' during the hours the attractions it covers (Colosseum/Forum/Palatine/etc.) are open but the actual ticketing portal is 'live' 24/7:
https://ticketing.colosseo.it/en
Your tickets to the colosseum will be date and entry time specific, and practically all of them will include the Forum and Palatine. To access SUPER Sites in the Forum requires a specific ticket or add-on to the 24-hr. general-entry ticket for the Colosseum/Forum/Palatine.
Adding to Kates response: I used the website she provided to get our tickets. However, you can not purchase tickets more than 30 days out. I had to wait until 30 days before the date we needed. But webste is very user friendly.
Thx guys. Also website for the Galleria Borghese Museum is stuck. Am I using the correct one?
https://galleriaborghese.beniculturali.it/
It's another website that's only 'live' during the hours the museum is open. But for 24/7 ticketing this their official site:
https://www.tosc.it/en/artist/galleria-borghese/galleria-borghese-2253937/?affiliate=T2C
FWIW, I really appreciated having the tickets with access to the SUPER sites in the Forum. If you can swing doing two separate days, one for the colosseum and one for the forum, I think that's smart too. It's great getting into the forum first thing when it opens and having the whole place to yourself.