Hi, and welcome to the forum!
I'm sorry that you're feeling stressed about your budget but take heart: Rome has umpty free things to do, and others which do not cost an arm and a leg. If it makes you feel better, we have yet to take any sort of guided tour of attractions in the city and done just fine. If you do want more help than just your guidebook (and you DO need a guidebook), some of them offer inexpensive rental audioguides, and Rick has some audioguides, including the Colosseum, which can be downloaded to your devices for free!
https://www.ricksteves.com/watch-read-listen/audio/audio-tours/italy
Let's start with Colosseum? No tours needed, although some folks with bigger budgets have chosen to take those. You can pre-order general-entry tickets from the website, those do not limit you to a particular day/time slot, and include the Forum and Palatine as well (those last two must be visited together; can't leave one and then come back for the other). Here's where you get tickets
(€ 12,00 apiece plus € 2,00 reservation fees).
http://ecm.coopculture.it/index.php?option=com_snapp&view=event&id=C1F4480E-9F88-41FE-BD39-1C2283FF1AF9&catalogid=9732C28F-DBBD-7AD3-39B3-015C8183692F&lang=en
They also offer 5€ guided tours of general areas of the arena that can be reserved by phone or online (if slots are still available). You will need to purchase general-entry tickets in addition to the tour:
http://ecm.coopculture.it/index.php?option=com_snapp&view=event&id=81BE288A-FD1E-48C9-9E06-4174F9CB7A74&catalogid=35195772-3F27-31ED-243B-015F57D668F9&lang=en
Something else to consider might be the Roma Passes: €38.50 for 72 consecutive hours, and entries to two of the attractions it covers will be "free". Others will be at a discount. It would also include your transport around Rome (but not to/from the airport) via buses and metro. As your hotel is near the end of the "B" metro line (Laurentina station) it'll take you some time to get in and out of the historical center; I doubt you'll want to be doing it RT more than once a day but the pass would cover the cost. Otherwise, individual BIT tickets are 1.5 euro. In your case, I think the pass might be a good deal, though, and you can purchase them when you get to Rome.
http://www.romapass.it
I don't know what's top of your wish list but the usual recommendation is to use the "free" entries at the Colosseum/Forum/Palatine and Galleria Borghese. One little problem with the Borghese is that you must make advance reservations BY PHONE even if using the pass, and you may be a little late in game for snagging open time slots but you can try. Otherwise, choose a different attraction, and preferably one of higher versus lower price.
"If you intend to visit the (Borghese) museum with your Roma Pass 72 Hours or Roma Pass 48 Hours, you must make your reservation on the phone, by calling +39 0632810."
Passes do not cover entry to the Vatican Museums or tours of other attractions...which leads to the next bonus: Roman churches. Almost all of them are free, oodles of them are glorious, and some are wonderful art museums in-and-of themselves. They are the budget-challenged dream! St Peter's, at the Vatican, is free but the museums - which include the Sistine - are not. They are also enormous, crowded and can eat up a ton of time. We prefer the Borghese for its smaller size and excellent crowd control but I know that the Sistine is high on a lot of first-timer's list. So, here's where you order general-entry, "open-tour" Vatican Museum tickets; € 17,00 euro + 4,00 reservation fee.
http://www.museivaticani.va/content/museivaticani/en/visita-i-musei/scegli-la-visita/musei-e-collezioni/musei-vaticani-e-cappella-sistina/visita-libera-musei-vaticani-e-cappella-sistina.html
Interesting free things (besides the hundreds of churches) include historic walks which may be laid out in your guidebook or an RS free download. (To be continued)