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 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][1] (as a "set") or [Electric, Motor 9V Power Functions M with Dark Bluish Gray Bottom][2] (as an "element")

Gears - multiple have been used in a video. You can find all of them in [Technic, Gear][3] section. These are used in pictured model:

 - [Technic, Gear 24 Tooth (2nd Version - 1 Axle Hole)][4]

 - [Technic, Gear Worm Screw, Long, Axle Opening Type II][5]

 - [Technic, Gearbox 2 x 4 x 3 1/3][6]

 - [Technic, Gear 40 Tooth][7]

Battery - Battery box used in video is [Electric 9V Battery Box Power Functions (Rechargeable) with Dark Bluish Gray Bottom][8]. 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][9] instead.

Please be advised both motors and battery boxes belong to [Power Functions][10] 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 Power Functions system. Each motor has a cable with connector, so no extra cables needed here.

    Is there a single kit that gets me 90% of the way there?

Pictured model doesn't use huge bunch of elements, so it rather hard to define what 90% is. Personally, I would probably look for some used sets with [motors][11] and [battery box][12]. And acquire the rest from Bricklink.


  [1]: https://www.bricklink.com/v2/catalog/catalogitem.page?S=8883-1
  [2]: https://www.bricklink.com/v2/catalog/catalogitem.page?P=58120c01
  [3]: https://www.bricklink.com/catalogList.asp?v=1&pg=1&catString=136&catType=P
  [4]: https://www.bricklink.com/v2/catalog/catalogitem.page?P=3648#T=C
  [5]: https://www.bricklink.com/v2/catalog/catalogitem.page?P=32905#T=C
  [6]: https://www.bricklink.com/v2/catalog/catalogitem.page?P=6588#T=C
  [7]: https://www.bricklink.com/v2/catalog/catalogitem.page?P=3649#T=C
  [8]: https://www.bricklink.com/v2/catalog/catalogitem.page?P=84599#T=C
  [9]: https://www.bricklink.com/v2/catalog/catalogitem.page?P=87513c01#T=C
  [10]: https://www.bricklink.com/v2/search.page?q=Power%20Functions#T=P
  [11]: https://www.bricklink.com/catalogItemIn.asp?P=58120c01&colorID=86&in=A
  [12]: https://www.bricklink.com/catalogItemIn.asp?P=87513c01&colorID=86&in=A