You can book entry tickets or tours of the Vatican museums and the colosseum online. Both will allow you to bypass the very long ticket lines.
For the Vatican museums you cannot use the bypass door into the basilica unless you're on a tour. They have become very strict with this and check tickets. If you take a tour just show your tour ticket and you can use the bypass.
https://biglietteriamusei.vatican.va/musei/tickets/do?weblang=en&do
For the colosseum, you can buy entry tickets at any time, they are not time nor date specific. This will allow you to bypass the long ticket lines. This ticket is valid for two consecutive days giving you a single entry into the colosseum, forum/palatine hill.
http://www.coopculture.it/colosseo-e-shop.cfm
You'll need to go through the security check point for the colosseum. This can get backed up and also the colosseum has a capacity limit which is often reached. When that happens they hold the lines until enough people leave. There is no way to tell when or if this will happen but just be aware that it can happen.
Donna