After the Ice Cube, I got my Rubik's 4x4x4 and my Rubik's 5x5x5, also known as the Rubik's Revenge (4x4x4) and the Rubik's Professor (5x5x5). These cubes were stiff at first, but after some lubrication to the 4x4x4, it turned much better. I average around 7 minutes for the 4x4x4 and my PB is 6:02.98. I've solved my 5x5x5 around four times. I don't know how fast I can solve one because I've never timed myself on it.
The 4x4x4's core is this type of "ball" core. The center pieces slide along a slim bar-like thing attached to the core. I don't know much about the 5x5x5's core, but I've heard it was kind of like the core of a 3x3x3 and has additional center and edge piece "slots".
The 4x4x4 is very stiff before lubrication but turns okay after lubrication. It does tend to "pop" when you cut corners or start really speed cubing with it. The 5x5x5 is also quite stiff out of the box. I have not lubed this, though. The 5x5x5 has this really weird turning. The outer layers are really stiff while the inner layers are looser. The 5x5x5 that I have has the ability to turn one separate inner layer without the outer layer turning, even when you don't hold on to the outer layer. Try this on your 3x3x3. It doesn't work, right? I really have no idea why this happens.
Overall, the Rubik's 4x4x4 and Rubik's 5x5x5 are pretty good after breaking in and lubing.