First thanks for hanging in there with linux. It can be difficult and sometimes even frustrating.
We were all beginners at some point. Having said that gentoo and knoppix are not exactly distros I would recommend for beginners. Maybe you want to check out the linux selection a little more? http://distrowatch.com/
There are some linux distros known for their ease of use, and they can prepare you to get more experiance so that if later you want to try something a little different you can.
At the link above there is a popularity column. At this time #'s 1, 2 & 4 are ease of use distros. There are others too, and you may find something that suits you better.
In knoppix you can enter "dmesg" at a terminal to see some of the start up messages.
you can also check in /var/log to look for various messages. Some of the files will be only accessible to root so you could use the file manager-super user mode to navigate to them and read them. (look for it in the kmenu under system)
I'm not sure exactly what you mean when you say you think you should be doing something after the system boots. "missing something" You can use "apt-get update" to make sure your source list is up to date. Don't do an upgrade! Not with knoppix. But you can "apt-get install (packagename) if you want a certain application. Good luck, hope this helps.