I have also come across lots of frustrations from rookie mistakes, and some of them took me time and thankfully helps from forum to figure out.
So this note is devoted to first-timers who are having troubles with the campaign, focusing on the scenario strategy and rookie mistakes I have made.
Hopefully you will find it useful, and entertaining from time to time.

It's partly based on Rommel level I'm currently playing, yet more than so often I had found myself in worse situations on other levels when I did not have so much of a clue of the game.
And it did put some ideas to the extreme for test, so I believe it's appropriate.
It's highly recommended, however, that you give it a try first before you read about regarded scenarios.
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0. Colonel or General?
I have no idea about the difficulty settings when I first saw them. It cost some do-overs.
Since I could not put it any better, I quote.
Colonel level is the default settings.Kerensky wrote:On Sergeant, all enemy units have a -5 strength modifier, some reduced AI ability, and the player has experience bonuses.
On Colonel, there are no strength or experience modifiers for each side, and the AI plays at an average level.
On Field Marshal, there are no strength modifiers, but the player receives a negative experience modifier, and the AI plays to the best of it's ability.
For anyone who thinks Field Marshal is still too easy... there's a surprise.
Many folks choose General level for strongest AI challenge while do not suffer from experience penalty.
While you will unlock three bonus difficulty levels after nailed Field Marshal, so it may not be a bad idea for advanced players to start with.
It should be considered a surprise, but since you have already come to the forum, I guess the odds are it's been ruined already. So I'm just going to go ahead and put it here.

Guderian level pushes you matching faster, for most the scenarios it's still reasonable, while achieving a decisive victory within 7 turns in the East Coast is too much. So I give it a pass.

Manstein level grants AI a huge strength bonus, so it's a lot of exciting battles.
Rommel level further halves your prestige, which means some strategic thinking, you have to plan for USA while in Poland.
Since the money is tight, it offers opportunities for better utilizing lesser units and unorthodox strategies.
So I picked Rommel, other than the fact the man is my favorite general of all time.
I thought Rommel was the easiest one of the three, for I had a 20,000-some prestige surplus after an earlier campaign victory, so how hard could it be?
Ooooh, how naive I was.

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1. Poland
I love this scenario a lot, very tutorly and unintimidating for starters. It's a warm-up for a grasp of some basic game concepts, very well designed.
Other than that, It's not much a challenge.
Only a couple of things worth mentioning,
1) Purchase extra core units asap, especially artillery, which helps cut the lost to minimal.
Usually I would go for 3 Sturmpanzers on turn 1, a humble goal I could not score before finished this scenario on Rommel.
2) Avoid reinforcing on battle fields, because in deploying phases you can have a 50% off for elite-reinforcing and free of charge for ordinary reinforcing.
It's something I used to pay heavily for.
As you may have noticed, in your army there is core units, which have a golden border, and auxiliary units, which have a silvery border.
Since you cannot carry auxiliary units over scenarios, Folks tend to leave them out of actions to maximize core units experiences.
Personally, I don't think it's necessary, for there seems to be a cap for XP, that is after some points for a certain scenario the amount of XP you will get for the same kill dramatically decreases.
However, the number of your kills does count and affects when you'll get heroes.
Anyway, it varies through difficulty levels, so better check out for yourself.

(Counter to common beliefs, Polish cavalry are a really threat to tanks.)
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2. Norway
It's the first deploying phase in the campaign, so we have a lot to cover.
1) SE units.
You might have received an SE unit from High Command at the very beginning of this scenario, if not, don't worry about it.
You could get up to three SE units throughout the whole campaign, infantries or tanks.
They are so well camouflaged that the almighty war-inspectors could not see them, so they do not take any unit slots, which means literally extra force and invaluable.
Though entitled "elite", they do not necessarily stronger than ordinary units of the same type, sometime even worse.
It's totally random events, and not affected by previous performance or how badly you need some extra force.
So it could be exploited by keeping loading the end of last scenario. I did so, for on Rommel an extra tank is crucial yet I would not have much use of more infantries.
Now thinking about it, I should have tried to sell them and see what happens, maybe High Command will send more, in a happy day dream

"When" you will get one does not as matters, however, and I would prefer getting them a bit later for possibly some advanced models.
2) Upgrading in series.
As a loyal customer, you can get a discount (usually only need to pay the difference) for upgrading in series. ( Pz3F -> Pz3G, Bf109E -> Bf109F...Usually the name gives you a clue)
And you can get a full refund by disbanding (hotkey "D") any unit, SE included, in deploying phase.
(NOT in the battle field, mind you, for then "D" stands for discard.)
Talking about consumerism.

3) Over-strength.
Experienced units could be over-strengthened, yet it charges sky high. Sadly not realizing this cost my first anabasis.
4) How to spend prestige.
Which is perhaps the most important question, and I still have not quite figured out yet.
Basically, prestige is spent on recruiting new units, upgrading them, and reinforcing them.
Ideally, you could keep max number of best units with most XP, yet usually you won't have near enough prestige for this especially as a first-timer.
It seems intuitive to keep elite-reinforcing to preserve XP. It'll cost you dearly, however, on average about 10~20 times a unit's original cost throughout the campaign in total.
It does not seem quite paid off. It's basically one extra initiative, attack, defence for every star, and so often you can get a much better deal from upgrading.
Number certainly matters. I'm not suggesting keep filling up every available slot, but a minimal number of units are required especially for a decisive victory.
More units mean better maneuverability and strategic possibility, and ganging up together helps.
For a previously attempt, I tried exclusively plain-reinforcing, and waltzed through Europe with easy and had more prestige than I could ever spend.
That is, however, until Midwest, which is hell.
On the other hand, challenging Midwest with lesser equipments also seems suicidal, and odds are most first-timers with prestige problems won't advance that far at all.
So I'm not sure which is more important, but it seems a very close call.
Still most players might prefer elite-reinforcing anyway, myself included. Just so you know, it's not the only option, at least for a part of your troop.

And for this scenario.
All in all, it's still a warming-up semi-tutorial. you are going to have your first landing operation and have to face some difficult terrain.
Maybe do some upgrading, but not necessarily right now. And I'd go for more artillery.
Before USA, I don't think I could have too many cannons. I believe I had 16~18 all sorts of big guns when I reached East Coast previously, and never regret any one of them.
In Panzer Corps artilleries come in with different rates of fire, so that big calibers are better against heavy armors and small ones are not totally useless.
I don't feel it's necessary for numbers and details here, just don't be surprised or enraged when you see some scary high-attack guns don't deal the damages you expected.
Some folks actually prefer a marginal victory in Norway, for you will get more prestige. And this applies for most part of the campaign as long as it does not affect the campaign path, for which you could check the in-game library.
In this case, you might want to skip the landing point near Trondheim Fjord on upper map. It's not very safe.
While aiming for decisive victory, it's pretty much a must. Some of our bravest navy guys sacrificing their lives for this opportunity, do not waste it.
Usually, I'd put two Sturmpanzers and one Gabirgsjager here, with two Fallschirmjagers joining force after they capture starting point airports and cities, keep advancing little by little until main force reach.
As for the other Fallschirmjager near Oslo, I find him much useful offering cover for landing troops, and after that, usually won't have much strength left to drop forward.
Terrain will slow you down, but you will still have plenty of time.
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3. Low Countries
It's a common turning around point for first-timers, myself included.
After getting some feels about the game, some decide better regroup for the long run, some feel too easy and go back to raise the challenge.
For me, I had some serious prestige crisis (High Command hated me), and could not pull it off the very first try.
So if you have doubts now, don't sweat on it, for I have had more than my fair shares of Low Countries.
The major issue here is previous prestige arrangement, as for the scenario itself, still very straightforward.
It's time to upgrade tanks and buy in some extra, one or two more fighters won't hurt either.
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4. France
After so many blitzkriegs, it's the first time the opponents can actually form a front line, so it's some exciting battles. But after that, there won't be much resistance.
Even on Rommel, I manage to build my artillery branch to 8 big guns. Couldn't hold back on Arts.
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5. Sea Lion 40
It's the first real challenge in the campaign. Without air and naval superiority, crossing the Channel is suicidal.
One of my earlier approach was to buy/upgrade some extra Fallschirmjagers and together with AUXs breaking through Lyme Regis and Ipswich, while main force concentrating on London and about.
It secures a quick decisive.
However, extra Fallschirmjagers are something I cannot afford on Rommel, so I chose to send AUX Fallschirmjagers assisting landing, mainly sacrificed.
Either way I would gather troop landing on Dover-Bexhill-on-sea line. It's much safer.
Two more fighters is necessary.
Besides that, I invested on two He-111 level bombers, it might be the only luxury I had through the entire campaign this time.
They are of great help at drawing fighter fires and sinking enemy ships. Besides, I figured I'd need them in East Coast anyway, might as well train them up early on.

Quick question: How comes London Held yet Paris fell in WW2?

Answer: London is 10 times bigger than Paris.