re: "wasn't aware you could do so much in it." After Effects is a huge Swiss army knife of a program, so yes, it handles animations, text, video, and special effects. You can use it to create simple animations with one element, but many complex TV show opens, ads, broadcast graphics and more are all made in After Effects.
I'll say that this is a pretty complex and ambitious look to use to start in After Effects. I would focus on a singular element to start, and see if you're able to wrap your head around things before you dive all the way in. Looking at this illustration, I see a lot of discrete pieces that would need to be created/animated, and then assembled together into a rig so they can move properly together, which can involve some of the more technical aspects of the program to be able to set up in a way that makes them easy to work with. Depending on the variety of animations you have in mind, part of me is inclined to agree with Rick that a proper 3D app might end up ultimately being more flexible for something like this, because you'd be able to set up a responsive rig that could be used for a variety of movements and angles. If everything will always be from this same or a similar view, than AE might be a good solution for you.
As far as the "rope," by which I'm assuming you mean the chains of carbohydrates/glycolipids: I would approach this by animating a stroked path and see if you are able to get the movement you're after. With no layers selected, use the Pen tool to create a line in the shape you'd like. You can enable animation on the Path property you just created, and animate it into the various shapes/movements. There are ways to attach other layers to this later (to get those little hexagons on there) but that's where I'd start!
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