What are the options for just doing a walk through on all our own? I
saw in Rick Steve's book that you can buy tickets in advance at
ticketclick.it - but I don't seem them on the site.
Pam, if it's the underground you're asking about, you can't access it at all without a tour. The 5-euro tours are for the main part of the arena above the underground, and are available to book only by phone: I covered that in a post above. However, you are not required to have a tour for that part and are perfectly free to visit it on your own. Some people do it with guidebooks, others with audioguides rented at the site, and others with Rick's free audioguide downloaded to their own devices.
Buy your general-entry tickets on the coopculture website: that's the OFFICIAL vendor. They are not day or time specific so you can go any day/time at all that the Colosseum, Forum and Palatine: (1 ticket covers all 3) are open before the end of this year.
http://www.coopculture.it/en/colosseo-e-shop.cfm
Adult tickets are € 12.00 +€ 2.00 reservation fee (€ 14.00).
Editing to add: I see you are traveling with two children ages 10 and 13? They will be "free" but you'll need to collect tickets for them at the special events counter. Choose the "pickup@ticket office" option when you do the checkout for your adult tickets. Bring your printed-out vouchers - along with your children's passports for proof of age - to the special events counter to collect all of your tickets: ask a guard if you are uncertain where to go.
You will all pass through a security checkpoint before proceeding to that counter.
(edited out/ corrected some info)