By "places to visit near the Colosseum" I'm going to assume you mean the Forum and Palatine Hill? If that's the case, I'll highly recommend a guided tour that includes all three (Colosseum/Forum/Palatine) as the latter two sites are more complex than than Colosseum and greatly benefit from an educated guide.
Parco Archeologico del Colosseo doesn't offer a tour which covers them all so you'll need to use an independent company, for example.....
https://theromanguy.com/tours/italy/rome/colosseum-dungeons-tour-with-roman-forum-palantine-hill?guests=1&_gl=1*1nw6tgd*_up*MQ..*_gs*MQ..&gclid=EAIaIQobChMIr92qhsyYjQMVGUT_AR1lSQ-rEAAYASAAEgJrvvD_BwE&gbraid=0AAAAADM-pak3vfnSP-TJKhQdBWsIy86CD
Some other tour companies people have used and recommended in the past have stopped including the underground portion of the Colosseum altogether in their tours as tickets were so difficult for them to get. I will call attention to the following under the "Frequently Asked Questions" section for the Roman Guy's tour linked above:
"Can my tour be canceled or moved to a different day?
Yes, it is possible. Colosseum Underground tours are very exclusive and are subject to changes, but we’re experts in this field and have staff dedicated to ensuring the best possible customer experience. If we are unable to acquire tickets on your behalf, we will let you know at least 14 days before your tour date, in which case we'll add you to a comparable tour or offer a full refund."
"Is it possible The Tour Guy will change this tour time?
Yes, it is very likely to happen due to the exclusive nature of this tour. We recommend you keep your entire day open to accommodate your new tour time. Be sure to monitor your email as we will contact you at least 14 days before the tour with any new start time."
You're not seeing FULL EXPERIENCE- UNDERGROUND LEVELS AND ARENA tickets on the Parco site as it's too early for bookings in the fall. That said - and again - they are very, very difficult to land, probably the most difficult of any attraction in Rome. Personally? As you can see down into large parts of it from above, the headache it is to obtain tickets from Parco wouldn't be worth the trouble for me.