There are several ways you can start with LEGO Technic.
If you have no elements whatsoever the best (cost-efficient) approach is to buy a set that contains most of the parts. Or a couple of sets. Bulk lots could also be useful, however, they may not contain the most expensive elements (like motors). Specific elements could be acquired from Bricklink or BrickOwl.
It is also cost-efficient to buy sets that contain electronic elements already since these are usually more expensive when bought individually.
Which motors, gearing systems, and batteries are they using here?
Motor - Power Functions M-Motor (as a "set") or Electric, Motor 9V Power Functions M with Dark Bluish Gray Bottom (as an "element")
Gears - multiple have been used in a video. You can find all of them in the Technic, Gear section. These are used in the pictured model:
Battery - The battery box used in the video is Electric 9V Battery Box Power Functions (Rechargeable) with Dark Bluish Gray Bottom. It is rather expensive and I suggest much cheaper and commonly found Electric 9V Battery Box Power Functions (Non-Rechargeable) with Dark Bluish Gray Bottom instead.
Please be advised both motors and battery boxes belong to the Power Functions system which has been recently discontinued by LEGO.
How are they connected to the SBrick?
SBrick is a controller and has connections compatible with the ones used by the Power Functions system. Each motor has a cable with a connector, so no extra cables are needed here.
Is there a single kit that gets me 90% of the way there?
The Pictured model doesn't use a huge bunch of elements, so it is rather hard to define what 90% is. Personally, I would probably look for some used sets with motors and battery box. And acquire the rest from Bricklink.
UPDATE: the second part of this video is adding a second adjustable joint to connect a 4th motor for climbing even more obstacles.
