Should I take the train from Civitavecchia to Rome and use the train to see all the major sites?

yep

Well, the Colosseum and the forum are right next to eachother but the colosseum is about 1.5km from the trevi fountain which you may want to walk to if you're up for it. If not, a taxi ride wouldn't cost you that much. Also, The trevi fountain, the Spanish steps and the Pantheon are all within walking distance of eachother and the best part is that there are lovely shops and coffee shops when you walk from the Pantheon to the trevi fountain and vice versa. The Vatican is quite far from all other attractions that you want to see so I suggest taking a taxi.