Well the thing I've found with this game, is that planes are lackluster until you get into the 1942 tech branch - and then they are (in high concentrations) *amazing*
I've watched TAC wings shatter divisions.
Anyhow..
So, Canada.
They aren't a good candidate for carriers simply because, to be effective, you need all the boat, navy doctrine, and air research from 3 different air related tech trees - which is a *lot* of crap to research.
However, Canada ended up with one of the largest navies in WW2 (after the French fleet was sunk by the British). What sort of ships you go for is up to you. With research as it is, going light cruiser/heavy cruiser is really viable (heavy cruiser for a bit of oomph + shore bombardment). But if you wanted a fancier navy you could probably afford a few BCs or BBs, it just depends on what you want. If you don't have a great deal of experience getting battleships to nations that ought not to have them, then at least for the first few attempts I'd stick to LC/CA or all DD (1to1 transport to DD ratio).
A light cruiser + CA/BC/BB or a DD + BB fleet would be great for Canada - a navy is wholly necessary, but the different ship types open you up to different strategic capabilities. Another interesting point to be made here, is that, particularly in the case of lighter ships (Lightcruisers or DDs whatever you pick - you won't be able to get both unless you forgo capital ships) you can buy production licenses for them typically. You could buy them from your allies as well - the British, French, and eventually the Americans should have pretty up-to-date DD/LC techs so it should be a pretty easy way to assuage some of the drain on your leadership if you can manage to secure a sufficient number of these production licenses and make some good ones earlier than normal.
From there, you have to sort out what sort of an army you're going to build. I'd opt for an elite force of infantry - a few MTN divisions some MAR divisions, PARA, and a few vanilla infantry to back them up. Because of your leadership situation, if you're going to build tanks or mot/cav/mech troops I would buy production licenses for such things rather than bother to research them - because you're simply not going to be able to make a tank that's comparable to the tanks fielded by majors without massively sacrificing either your industrial techs or your naval techs (which are somewhat essential, since without the naval techs you can't safely transport the troops you'll spend your IC on).
Depending on your ambitions for Canada, you can also try to get a bigger army through production licenses and tech stealing (from France and UK). For your Air Force, you're basically only going to have enough to research one kind of craft - I'd pry go with INT just for their general purpose nature (I mean you can actually bomb with them too - just won't be doing tons of damage). If you can manage to secure the money, you could buy production licenses for the planes and get pretty decent ones (you'll have to sell a lot of supplies and fuel) until your theories (particularly combat experience) get a little higher, to make it economically viable.
I'd still build IC as Canada for a bit regardless. You'll have to wait on the tech - so just build air bases until it finishes. Building even a simple DD only fleet will absorb a lot of your IC for quite some time.
Once you've got Canada ready for war, if you join the allies I'd send everything I had to France. The French AI will still likely lose the war no matter what sort of help you'll give them, but you'll be able to sex up some of your generals with more desirable traits, get some EXP on your divisions, and get that all important land combat experience, and maybe work in a little convoy raiding (with the home fleet nearby you should be pretty OK). IRL the Canadians actually made a landing into France *after* they had surrendered, because apparently somebody had forgot to inform them of France's surrender, so they accidentally landed in France and successfully liberated 1 small coastal town before being informed of the situation and quickly departing. If you want to really lame it up, you could also race to Europe and get yourself mixed up in some fights for easy EXP. A few Maginot crossings
Make sure you can get whatever troops you have in France *out* of France again if they fall on their own sword.
Once you've dealt with the battle for France, likely losing, but hey, maybe things can work out. Africa would be your next likely target. Particularly Somalia and Ethiopia. With the right kinds of elite infantry, or maybe even PARAs these should go down *very* quickly, and the red sea fleet of Italy shouldn't be able to contest you no matter what sort of a rust-bucket fleet you've built lol.
Now Canada has some real punching power

An Imperial power!
After that you can likely focus on Africa for a good long while - get yourself Libya, maybe Vichy Africa, Syria, etc. You could also mess around in Norway (MTN troops) to get some more elite divisions and generals, and distract Germany.
Also Persia may go Axis (particularly if Italy goes ham in Africa) which opens up the possibility to puppet them...
Then you'll be all set to invade Italy

Italy makes the best landing choice for Canada because you'll never have many divisions, *but* with an elite and experienced force of MTN brigades, and marines for your flanks + stock infantry + maybe PARAs (you could use your new bases in Libya) you'd be able to get quite the foothold in Corsica, Sardinia, and Sicily. Maybe even Crete. Even if the USSR completely screws the pooch and loses, you should be able to hold out here for a pretty long while. The Italian navy is formidable, but you have the British fleet to help you in most cases so it shouldn't be too worrisome, plus the majority of the AI Axis' forces will be in the USSR, so resistance may end up being fairly light!
Once you've set up the West to your satisfaction - it'll be time to go to Asia.
See all those Yellow colored Islands? Time to paint them RED! The Japanese fleet is formidable, so you may want to wait until you see lots of US fleets scurrying around the pacific before you make your move unless you've managed to construct a glorious battle fleet. Then just Island hop your way for imperial glory, until you get in range of the biggest yellow colored Islands. Namely Taiwan, Hokkaido, Toyohara, Shikoku, and Kyushu. Force concentrations on these islands should be fairly low (most of Japan's army is typically locked up in mainland Asia, various islands, and sometimes Australia by this point). Also it's incredibly hard to cross the straits, so you should be able to get in and lock down the islands before the more substantial forces on the big island can respond. Then you just bide your time until you can get 15-20 divisions to to mainland Japan, and then you truly are a world power

.
Depending on if you'd like to be really ahistorical and gamey, you could also look at knocking over a few Caribbean nations (once the US goes Allies), Portugal, Persia, Siam, and basically any other smallerish nations with a coast line. If the USSR doesn't collapse you may also be able to sneak attack Bulgaria and *maybe* Romania if you were EXTREMELY lucky once the Axis starts to collapse - ensuring Canadian sponsored "Democracy" in the Balkans.