The 4D sphere isn't all that hard. Here's how you get a mental model of it:
1) Take a 2D circle. But don't imagine the overhead view of the circle that we usually think of. Instead, think of it as seen edge-on. Or, if you like, look at a cylinder sideways. Notice the "bulge" in the middle, where it curves "outwards" at you? Now imagine the cylinder becoming flatter and flatter. Keep that "bulge" in the middle in mind. Imagine the cylinder as flat as a razor-thin strip. Now you have a line segment, but it's not really a line segment; it's an edge-on circle.
2) Take the 3D sphere. As seen on your computer screen, it's actually nothing more than a shaded circle. But actually, it's not a circle; the middle "bulges out", as conveyed by the way it's shaded. Notice how the envelope of the sphere is just a circle on your 2D screen, but your mind "sees" it as a sphere with a bulge in the middle, not a flat circle.
3) Now take the 4D sphere. In projection it's just a 3D sphere. However, it isn't a "flat" 3D sphere; it has a "bulge" in the middle, i.e., inside the core of the spherical projection image. The part of the 4D sphere's surface that projects to the inside is "bulging out" at you; it's nearer to you than its spherical boundary. Now you've "seen" the 4D sphere.
Easy, wasn't it?

Of course, that was only half of it. The "bulging out" part is the half of the 4D sphere facing you. Just like when you look at a 3D sphere, the "bulging out" is the near half. There's another half on the far side, that, if you cut away the near half, also occupies the space of a circular disk, but it "caves in" rather than bulges out. Glue these two halves together in your mind, and you've a complete picture of the 3D sphere.
Now do the same thing with the 4D sphere: as described above, it's just a 3D sphere, or rather ball (filled sphere), where the insides are "bulging out" at you. Well this is actually only half of the 4D sphere. The other half lies on the "far side", also spherical in shape, but ths insides are "caving in", i.e., curved away from you in the 4th direction. Glue these two halves together, and there you have it: you've successfully visualized the entire 4D sphere.
