I guess mine is somewhat similar to what Jane Monroe answered. I play the Koreans so I first upgraded the heavy singijeon. Then I researched the cannon followed by supply train. My 4th research was the wall sapper, but I recommend researching cuirassier before you research wall sapper. In short, I've done these upgrades in order: Heavy singijeon (musketeers), cannon, supply train, wall sapper, then cuirassier. My next upgrade will be fusilier and I doubt I'll bother researching carabineer raider since I never use them. Really the most important troops in the game is ranged infantry, siege (what cannon/bombard/artillery are called), health cart, and heavy cavalry. I used to use heavy infantry troops a lot when I played the Japanese nation, but ever since I switched to the Koreans I've stopped using them. Heavy infantry aren't really needed IMO especially in the EA and IA because they get killed too easily and quickly. Ranged infantry are much better troops and can do damage from a distance. That doesn't mean you shouldn't research ashigaru (I know I will research fusilier, but I reserve it for last since I won't research EA raiders). I used to bring a wall sapper with me to every battle, but since I extremely rarely use it it's no longer a part of my army.
Basically I would research musketeers, cannon, supply wagon (then get supply train ASAP) and cuirassier in that order. I suppose you could research ashigaru first since it's your unique unit and if you use it, but a lot of players who play the Germans or Japanese in the EA or IA don't use the unique heavy infantry, so it's up to you. I just don't think heavy infantry units are important enough to research first, second, third, or even fourth.