As no one else has mentioned it, progress towards this achievement can be tracked by going to 'Menu' in the bottom left corner of the main menu, then clicking 'Showcase' and then 'Achievements'. This shows all of the in-game achievements which incidentally are also Steam Achievements on the Steam version, however only four of them are Xbox achievements (we got off lucky). Progress tracked on these does in my experience correlate to your actual achievement progress.
Conscripts: can only be trained from HQ while playing as the Soviet Union. I am not 100% sure but I do not think call ins (Fresh Conscripts, Frontoviki Conscripts, Penal Battalions) count for this as all have different unit names from the generic Conscript Squads and are not trained but rather called in instantly.
In any case, I did all of my conscripts in one playthrough of the campaign on General (actually called 'Hard' in game). The campaign is pretty easy all said and done as the AI normally just defends in most missions and you can just keep building more and more soldiers to replenish losses (with some exceptions like the mission with Polish Partisans and the Poznan (Posen) mission where if you take too long, you lose the ability to train any more troops).
Conscripts are pretty versatile anyway as can be used to replenish losses in better squads such as Shock Troops and Guards as well as being used to man crew served weapons such as AT guns, SF-role machine guns (i.e. ones that must be crewed rather than be assigned to a squad member) and to crew captured vehicles. As such just pump them out whenever you have 6 Population and 240 Manpower to spare and you'll get this without ever touching Skirmish as Soviets.
Grenadiers: Can only be trained by the 'Wehrmacht' faction. Note there are two Wehrmacht factions. 'Wehrmacht' (usually referred to in CoH community as 'Wehrmacht Ostheer' to differentiate) is the one you want as opposed to Oberkommando West who train Volksgrenadiers instead of Grenadiers. These are trained from the Infanterie Kompanie which can be constructed by your Pioneers as soon as a match begins and without reaching the next Battle Phase at the HQ. They cost 240 Manpower and take up 7 Population.
Panzergrenadiers: can again only be trained by Wehrmacht Ostheer. These are trained at the HQ building but require Battle Phase 1 to be reached at the HQ. They cost 340 Manpower and 9 Population.
For these latter two, I generally just started a game against the AI on easy, churned out a few of each until the manpower generation started to drop to a crawl whilst capturing all of the zones and killing off enemy infantry. You can box the enemy into their base by positioning all your infantry in the cap zones around their base. PzG units wreck most infantry squads especially when supported by Grenadiers with an MG42 upgrade. Playing against Oberkommando West or US Forces seems to make this process marginally easier as their infantry isn't great and they seem to train a lot of light vehicles which can be taken out by infantry armed with small arms (British are annoying as they will constantly drop air landing glider units behind your lines). Once the manpower generation isn't worth the wait, just train an armoured unit (I normally just went for PzIV as it's cheap but effective) and roll into the enemy base supported by infantry. Not sure if this is actually necessary as opposed to quitting but I did it to make sure the game ended and saved my stats correctly.