That entire tutorial was one huge chuckle-fest... Who knows what they were smoking when they made that.

The poor moustachioed man going to shoot himself...
Anyway; the tutorial does a reasonable job of explaining the very basics. But there are some things that are still eluding me. Which is a good nation to play to get to grips with? Should I pick one right at the heart of the action like Germany, or a more obscure nation like Canada? With my IC points, which of the sliders is more important to invest most in? I'd say the upgrade slider is most useful but I'm also thinking it will vary depending on the situation. Could I get away with just giving the lowest recommended amounts of consumer goods, to put more effort into my war machine?
You should always keep just enough points in Consumer Goods to keep your Dissent at zero. The only advantage to producing extra CGs is that it will reduce any Dissent you might have.
I started off playing Germany, since they are strong enough that you can afford to make mistakes. You could try the September 1939 scenario first, which pitches you straight into the war with Poland. It will teach you how combat works. Once you've beaten Poland, you can re-start a new game with the 1936 scenario, and learn research, production, diplomacy, etc.
The USA is also a good learning game, since you are nearly immune to invasion. Not much will happen for the first few years, though, since you are so neutral that most of your production is required in Consumer Goods to keep your population content. As your neutrality drops, more and more production can go into preparing for war.
If you are at peace, upgrades should be deferred as long as possible, as long as you get them done before sending your men into battle. This is because there is an upgrade cost-and-time discount for units that are more than one step obsolete. The unit will quickly upgrade until it is only one step obsolete, then more slowly until it is fully up-to-date... so upgrading it every time it drops behind the curve is more effective (during wartime) but also more expensive (during peacetime).
EDIT:
One of the biggest differences between HOI-3 and other wargames (such as EU-3) is the concept of
Movement Is Attack. As soon as you begin moving into an enemy-controlled province, the battle starts; and the combat will finish before your unit actually reaches the province. There will then be a cooling-off period during which the unit will not respond to orders... it is reorganizing, clearing away casualties, bringing up replacements, adusting its supply lines, etc. It might be two or three days before the unit will be ready for new orders.
Another difference is the way that HOI-3 handles stacking limits. EU-3 has no stacking limit, but penalizes large stacks with attrition. Some war-games have hard stacking caps, allowing only a certain number of units to enter a province. HOI-3 handles this matter differently, with Frontage and Combat Width.
Every province-to-province boundary has a certain Frontage... ten Frontage for the first adjacent province, plus five more for each additional province that you are attacking from. So if you were attacking from three provinces into one enemy province, your total Frontage would be twenty (10+5+5).
Combat units have a certain Combat Width... typically one width per line regiment, zero per support regiment, and two per pre-'Spearhead' Armored Regiment (there is a tech you can study that reduces the Combat Width of Armor to the same as that of Infantry).
So a Division composed of three Infantry regiments and one Artillery regiment would have a combat width of three (the ART is a support regiment, width zero), while an early-war Division composed of two Armored regiments, one Motorized Infanty regiment and one Self-Propelled Artillery regiment would have a width of five (two for each pre-'Spearhead' ARM, one for the MOT, zero for the SPA).
You can continue adding units to a battle until you have
exceeded (not equalled) the frontage... so for that three-province, twenty-frontage attack mentioned earlier, you could use seven INFx3/ARTx1 Divisions (width 7x3=21) or five ARMx2/MOTx1/SPAx1 Divisions (width 5x5=25).
There are also stacking modifiers, unrelated to frontage, which penalize players who use too many different formations in the same battle. Without penalty, you can attack with one unit plus three more per attack direction... so a one-province attack can use four units (1+3) without penalty; a two-province attack can use seven (1+3+3); a three-province attack can use ten (1+3+3+3); and so on. Extra units beyond the stacking limit will penalize your entire attack (not just the over-stacked units... everybody) by -10% multiplicative per extra unit... so if you are three units over the limit, you are penalized by -27.9% (ie: 90% of 90% of 90% = 72.1%). Your Theatre commander uses his skill bonus to reduce these penalties.