As for the 600-cell, you can use the same layers method to visualize it. Start with a single point, call it the North Pole, and then take 20 tetrahedra and attach them to this point in the geometry of an icosahedron. In 3D, this leaves some tiny gaps between them; this gives you a bit of room to fold them slightly into 4D so that the gaps close up, and you get a sorta 4D "floret" with an icosahedral boundary. This is the 1st layer.
Then you stick on another 20 tetrahedra to this icosahedral floret, which gives you a spiky bowl which you can fit 30 tetrahedra into (corresponding to the edges of the icosahedron). This gives you a sorta 4D bowl with 12 pentagonal gaps; each gap can take 5 tetrahedra. You may consider this as the "2nd layer", in a sense. The "rim" of this bowl is now a pentakis dodecahedron (a dodecahedron with 12 pentagonal pyramids stuck to its faces).
From here, it gets really hard to describe the next layer in words... you take 30 pairs of tetrahedra and stick them on so that they straddle the edges where the pentagonal pyramids touch. You then get grooves radiating from the tips of the pentagonal pyramids, going between each of these pairs. Since there are 5 such grooves per pentagonal pyramid, and there are 12 pyramids, you stick on another 5*12 = 60 tetrahedra to fill in these grooves. Now you end up with a 4D bowl where the "rim" has 20 conspicuous tetrahedron-shaped gaps, so you fill in the 20 tetrahedra to complete the 3rd layer.
Now take two copies of these 3 layers, and join them together where you filled in the last 20 tetrahedra. However, just as with the 120-cell, the other 3rd layer cells from each bowl don't join up with each other; they leave 12 gaps shaped like flat pentagonal bipyramids: each of these gaps are filled up by 5 tetrahedra -- so that's another 60 tetrahedra to fill in the "equator".
If you add everything up, you'll find that there are 600 cells.
And yes, most of this probably went right over your head because this is really hard to describe in words, so I'll just have to shamelessly point out my website again, where you get to see the pictures that hopefully are worth 1000 words:
http://eusebeia.dyndns.org/4d/600-cell.htmlNow, mind you, there are other, less complex ways to derive the 600-cell, but they involve geometric operations that aren't easy to visualize, so I personally didn't find it very helpful until after the fact. One way is to take a 24-cell, divide its edges in the Golden Ratio in such a way that cutting the 24-cell down to those edges gives you regular icosahedra. (See, I told you it wasn't easy to visualize.) This process introduces a bunch of new tetrahedral cells, so you end up with something called the "snub 24-cell", which has 24 icosahedral cells and a whole bunch of tetrahedra joining them (120 of them, to be precise). Now take 480 tetrahedra, and form them into 24 icosahedral pyramids (just like the 1st layer I described above), and stick these pyramids onto the icosahedral cells of the snub 24-cell. Bingo! You get the 600-cell.
Yeah, that's not really that much easier to understand.
The best way to get the 600-cell is just to take the dual of the 120-cell: take a 120-cell, find the center of all 120 dodecahedra, discard the dodecahedra but keep the centers, now wrap some 4D shrink-wrap around all 120 points and shrink-wrap them. Your 4D shrinkwrap is now in the shape of a 600-cell, brand new, shrinkwrapped.