With the new patch 2.0.1, I'm level 60 and I can't get to level 61. I keep getting paragon levels, but I can't get "normal" levels to get to level 70. How can I gain "normal" XP to actually level up?
Level 70 is tied to the upcoming expansion, Reaper of Souls (as is the Crusader class and a few other things).
Many of the rest of the changes were released in patch 2.0.1, and while it was released in preparation for the expansion, they are not related, and everyone who owns the base game will be able to enjoy their benefits (as you and your friends have experienced).
World of Warcraft (a small Blizzard game you may have heard of) works in a similar way, with a "pre-expansion patch" that contains the major system changes that were decided to be released alongside the expansion.
I think you have to have the Reaper of Souls expansion either installed or at least linked to you account to get to level 70. After Paragon 2.0 goes live, if you are level 60 and do not have the Reaper of Souls enabled on your account, you will continue to earn Paragon experience in the new system. This is because you are still technically at the level cap. However, if you are level 60 and decide to enable the expansion on your account, you will stop earning Paragon experience. This is because the expansion increases the level cap to 70. Once you reach level 70, you will start to accrue Paragon experience again from where you left off. (So, basically: Only max-level characters will be able to earn Paragon experience. If you don't have the expansion, max-level = 60. If you have the expansion, max-level = 70.)