To join the individual developer program you first buy the entry from the Apple online store at store.apple.com (perhaps in your local country's store if you don't work out of or live in the USA.)
From there, you pay with whatever Apple ID you want since you can choose later to join with that or another Apple ID in the developer portal called iTunes Connect.
You have two sources of support while buying any paid developer account - the first is web and phone sales from http://store.apple.com - you can and should call them and explain your difficulties and they will help you with your purchase. If they can't resolve the issue they may refer you to AppleCare for AppleID issues and/or Apple Developer Support for specific issues on the developer program:
As to what's happening, it's hard to guess - but if you don't have an AppleID that works to log into the online store - work that out first and make the purchase. Once you've spent money on the purchase, you will have a very narrow list of things to check/correct with the help of support if needed. If you can't get an Apple ID to work for the initial purchase, that too is pretty easy to sort with Apple's phone or online support. Worst case, ask a second question here with exact URL you are using and details on which URL you can't get logged in to.