Unexpectedly persistent problems with connecting to Store

This has been going on for months, maybe a whole year, didn't pay much attention to it because I don't use a lot of apps on my iPhone and thought the problem had something to do with my setup getting dangerously close to obsolescence, at the time I was running Mac OS X 10.6.8 on a 2007 iMac and iOS 6.1.3 on an iPhone 4.

One day, out of the blue, I couldn't connect to the App Store via iTunes, all I got when opening the Store tab was a blank page, and by blank I mean completely empty, white... I could still access updates of the apps I already had, and I could get to single apps in the Store via a search, but when I tried to download anything new the requester for my Apple ID and password just kept popping up in a loop, never getting anywhere.

I recently changed my whole setup, brand new Mac Mini running OS X 10.10.2, iTunes 12.1.0.50, but the problem didn't disappear as I had hoped it would, it got worse instead, everything's the same AND I can't even get updates for apps I already have, the "looped" requester pops up for those, too...

I have to point out that my Apple ID and password are correct and functioning everywhere else, as well as my credit card info is up to date, I bought the very computer I'm using on the Apple Store a couple of months ago, had no problem accessing the App Store for System updates, iTunes actually greets me by first name when I launch it and, of course, I'm writing here, so it DOES work...

Can anyone understand why this is happening to me and, hopefully, know how to fix it because, although my iPhone is fairly secondary in my informatic life, still the same and with same, old iOS, all this is driving me barmy...

Cheers!