The MAIN PANEL contains the first disconnect past the meter. The ground rod line must connect here (to the ground bar if they are separated). Also, here is the location of the neutral-ground equipotential bond tying neutrals and grounds together.
The panel you linked has four little NEUTRAL bars
up high near the top of the panel. Your ACTUAL panel has three little NEUTRAl bars up high near the top of the panel. Note that all these are insulated from the panel chassis, that's what makes them neutral bars and not ground bars. So clearly your panel is NOT the one you linked. I suspect it is a 24/24 (i.e. allows zero tandem breakers).
If you had bars
down low near the bottom of the panel, those would be optional, added GROUND bars. They would not be insulated from the chassis. Separating neutrals and grounds is a) mandatory in a subpanel. b) optional, but elegant and well-done in a main panel.
In a MAIN panel, grounds are allowed as guests on the neutral bar. Cheap builders rely on that a lot.
This panel can be configured either as a main panel or a subpanel.
I noticed that he put all of the neutrals on the upper bars on both sides and the grounds on the lower bars on both sides. Should the neutrals and grounds be separated like that on the two different bars?
On your panel, the "bars near the top of the panel" are staircased a little bit for easy wire entry. You seem to be calling that "upper" and "lower". That is wasted, they are all neutral bars. They are all insulated from the panel. Your panel according to your photo has no accessory ground bars. Feel free to add some. The panel labeling lists specific model numbers of ground bar designed to fit pre-drilled, pre-tapped sites all over the panel.
If this is a main panel, it doesn't matter. Separation is mandatory on a subpanel.
Note that in a main panel, you still have that neutral-ground equipotential bond.
And should there be a ground off the lower neutral bar to the grounding rod?
YES. The Grounding Electrode System (ground rod and connecting wires) connects to the GROUND bar if we're in a place where that matters.
Part of me doesn’t think it’s correct, the reason why is this panel has 24 spaces, if the neutrals are on the two top bars and grounds on the lower two bars there isn’t enough room for 24 breakers?
Well, given that you linked the wrong panel, I suspect your panel isn't actually a 24/34, it's a 24/24. There have to be enough neutral positions, or UL won't give the panel a UL Listing.
However, this is also a Plug-On Neutral (PON) panel designed to allow AFCI and GFCI breakers to pick up neutral by the vertical rails instead of taking a space on the neutral bar. Food for thought.
UL requires each neutral must be alone in its slot.
Would I have to double up some of the neutrals and grounds or should there be one neutral and one ground together in one slot?
NEC 408.21 [Neutral] Terminations. Each [neutral] shell terminate within the panelboard in an individual terminal that is not also used for another conductor.
However this has been a UL requirement for >50 years, just people weren't reading the instructions, so they had to put it into NEC.
Especially do not put the circuit's ground next to the neutral. If that had a connection problem, it could electrify all the grounds on the circuit!
However, GROUNDS are a different deal. Read the panel labeling, it often allows multiple grounds per slot.
I have read that you can put one neutral and one ground together only if it’s in the main panel.
Throw that book in the trash. It's horsepuckey. As you plainly see, neutrals must be alone.
Now it is true that grounds are allowed as guests on the neutral bar, but they share a spot, they can only share with other grounds, and to the limits specified in the panel labeling.