Ah you will love Rome! It's literally like an open-air museum. I went in June and I hope my tips etc. help you a little!
There are two free English tours you can do, if you google 'Rome free walking tour', one goes to the Colosseum and other things along the way and the other goes to the Pantheon (amazing building) and on to the outside of the Vatican (it ends at the Vatican). They do the tours every day from the Spanish steps (meeting place), one at 10am and one at 4pm. Take comfy shoes because there is a LOT of walking, but it's really cool!
Then what we did when we got to the Vatican, we went into the church there (Basilica), where we paid to go to the dome at the top. The view was incredible! You have to walk up 500+ steps though, haha. The staircases start off straight and then they're spiralled! It's crazy. Be aware of the dress code in religious buildings, too. Women and men are refused entry in some places if their shoulders or knees aren't covered. You have to dress appropriately (google it if you wanna know more)...they sent some people away.
Then we booked the Vatican museum online in advance (otherwise the queue is absolutely huge, but you can skip the most of it if you book online). It's huge - SO much to see, but it's cool! You get to see the Sistine chapel in there, too. It can be a bit much because there is so much to see, but I enjoyed it. It took us a whole day.
If you like gelato, go to Gelateria Del Teatro! They do the best gelato ever! They have crazy flavours like wine and chocolate, peach and lavender, white chocolate and basil...and then all the normal ones. I would definitely recommend it!!
The Trevi fountain is cool but very crowded - a must see though!
Then we had a day trip on the train outside the city to these ruins, which was also incredible. The ruin place was called Ostia Antica - you can get the train there, and get a ticket and it's a huge walk through these ruins - so much to see! A lot of it is quite similar, but there are some awesome hidden gems there! Comfy shoes are a must again - it's very uneven everywhere!
The food is also amazing - if you want any restaurant recommendations, just let me know! Oh and be careful when going to some restaurants if you get a panini for lunch or something and sit down there, some of the crafty buggers charge you extra for sitting in, hehe. Have an amazing time!