Leveling up Production Buildings on Day 2

Your instinct matches my approach as well. It's often of little value to level up your troop productions buildings until later in the game. This is true for two reasons. First, until your economy can support keeping your existing troop production buildings in constant operation, it's of no value to upgrade them. If you spend the resources upgrading the building, that's less resources you'll have to actually build those type of troops. Essentially this ends up being counter productive as the buildings often require the same resources as the troops themselves. The second reason is that each unit has a minimum production time, so if you're already at this minimum time, upgrading the buildings will be of no value. As a general rule your level of troop production building should match the level of unit you are building. This will vary slightly if you are building advanced units like Motorized Infantry or any SP unit.

Yea, you only need to level up production buildings when speed of production no longer keeps up with your resource generation. Until then, you're fine with level 1 buildings. I usually jump my units from level 1 to level 3, and only then start upgrading my production buildings, so maybe day 5 or 6.