Colesseum

You can purchase online here.

https://www.coopculture.it/en/colosseo-e-shop.cfm

As far as the Rome Pass, it does not include the upper and lower levels.
Those can be purchased separately on the above site. The pass may make sense if you do 2 major sites plus a reduced site and plan on using a few bus or metro rides.
You have to do the math to decide.

An example would be the Colosseum/Forum/Palatine Hill, (counted as 1 but with 2/days to do them.) Borghese Gallery, and a 1/2 priced ticket to the Capitoline Museum. Plus 2-4 rides on transportation.

If anything you need to do the opposite of what you were thinking.

Should we just reserve regular tickets now and add on underground and top tier when we get there?
The regular tickets do not sell out BUT the underground and top tier do ; if you wait until you get there you will not be able to buy those.

Correct! Buy your underground Colosseum tickets as soon as they become available. They look to be available about one month out.
Buy your Roma Pass when you get to Rome. It is usually available at a tobacco store or TI. It will let you use a special entrance with no waiting in line at the Colosseum.

Not only do they sell out, but they sell out quickly. There are a ton of threads about buying the tickets on the morning that they become available. Don't know how much it has changed since they separated the top tier and the underground but I would assume it is just as fast (my date was sold out within hours). And remember no matter what there is a line to get through security - that alone can took about 40 minutes to the Colosseum. When we crossed over to the Palatine Hill it was another 15 minutes to get through security there.

Not only do they sell out, but they sell out quickly.

True. For high/shoulder season, they usually sell out in minutes after they become available online.

If you want both the underground and upper tiers, this is the tour to purchase:

http://ecm.coopculture.it/index.php?option=com_snapp&view=event&id=E70BA522-3BA1-4BC5-071A-015FBFEA7456&catalogid=AC297F8A-566C-CF3B-18A5-015FC0002E80&lang=en

It's € 15,00 + a € 2,00 reservation fee, and note that you must purchase a general-entry ticket IN ADDITION to the tour. You can purchase that ticket at the events desk when you arrive for your tour.

Tickets for the Underground and/or Belvedere go on the sale on the third Monday of each month for the following month at 9:00 AM ROME TIME, unless that particular Monday is a holiday. For instance, here was the website (coopculture) notice for Feb. tickets:

"Opening of reservations for Undergrounds of Colosseum and Belvedere for February 2018
We inform you that on January 15th 2018 it will be possible to book the undergrounds and Belvedere tours for the month of February 2018. Any online bookings made before 9.00 am will be not valid."

So, tickets for March should become available on Feb. 19th. For April, on March 19, etc.

Editing to add: your general-entry tickets include entry to the Forum+ Palatine.

Thank you everyone. So the best thing is to book underground as soon as they go on sale for June visit and buy Roma Pass as well. We can use the Roma Pass and pre booked underground tour to skip lines?David

Much is made about the lines and skipping the lines at the Colesseum but it is not as great a time savings as you might think.
Sometimes yes if the line to buy tickets is long but you still have security to go through so a different line which is still long.
There are 2 lines one for un-ticketed guests and one for guests with tickets or passes.

Last time I was there Summer of 2017 the line with Roma Pass or pre-purchased tickets was approx. 30 minutes and the line with no tickets at all was approx. 45 minutes so you have a long line that is unavoidable most likely for your time in June.

Thinking back to our previous time there a few years back (2011) I recall the pre-bought ticket line for security was much shorter and moved faster so likely a combo of more pre-bought tickets thanks to the Underground tours as well as more reading about it as well as overall increased security. In 2011 the underground area was not yet open so not available ; it was just after our trip the same year that they first opened it and offered the tours.

Agreeing with mreynolds: the tour and Roma pass will let you skip the loooong ticket line but not the security check queue...and that can be lengthy. I'm seeing recommendations to arrive at least 1 hour before your tour slot to make sure you get through the security queue and to the event desk to check in on time.

As the website states:
"Please note that for safety reasons the Colosseum can accommodate up to 3.000 people at once. This could lead to delays in access to the site, even for pre-booked visitors."

As Rome's #1 attraction, I can't overstate how busy it is, and especially so during high and shoulder seasons. Give yourself PLENTY of time.