PROBLEM SOLVED.
I solved my own problem and had fun doing it.
Went to You Tube and found a video that describes a partial solution to the "iTunes not connecting" and getting "Network timeouts"
They recommened using the "NORTON REMOVAL TOOL".
Prior to applying the "Norton Removal Tool", I performed the following on my HP laptop with Windows 8 installed.
Reason I tried tried the following ipconfig commands is because It appeared that the DNS did not recognize the iTunes Store address.
using the DOS CMD console as administrator,
ipconfig/release
ipconfig/flushdns
ipconfig/renew
ipconfig/all
Ethernet addresses didn't appear to change.
Retried connecting to iTunesStore, no luck.
Tried the next thing , because the person fixed the same problem I had, so why not try it on my machine.
again using the DOS CMD console as administrator,
netsh winsock reset
received a message back that I needed to restart the computer.
After everything was back up and log in to my account. Tried to get to iTunes Store again, no luck.
This when I tried the "Norton Removal Tool".
After removal of any and all related Norton apps, I tried to get to iTune Store, and "wa laa" ( a term used by magicians), I connected to iTunes Store and iUniversity and a whole bunch of other stuff.
This worked for one laptop computer. I also did this for my other laptop computer.
I Hope I don't run out of space here.
To make a long story short, you need to run everything I ran above, because I couldn't get the other 2 computers to work unless I ran all the above commands and ran the Norton Removal Tool.
Yes, I know what you're thinking. I was able to all the computers to run with Norton 360 reinstalled. NO MORE PROBLEMS with iTunes Store Connection.
Enjoy as I spent way too much time on this problem.