Your aim is to set up a one-on-one chat similar to a phone call, with limited experience. That is what skype is designed to do; you will need to register, but after that, the rest should be dead simple.
The number skype asks for, I believe, is in case you want others to be able to call with a landline to reach you on skype. This is not necessary for your use as described, and you should be able to ignore that. It will also offer you various subscriptions (voicemail etc). Ignore those as well, unless you find that is something you'll want. The basic skype service - calls from PC A to PC B - is free.
It will say you have a free call you can make, but don't be confused - that is a free call to a land phone. All PC-to-PC conversations are 100% free. (Aside from the internet connection).
Skype has become a little bloated, but it's still one of, if not the, best VOIP program for one-on-one or (very) small group conversations.