Should I be advancing to the next Age/Level of units immediately or max out on the current Age first?

I am very much interested in strategy games such as Age of Empires, and one thing I always find hard to figure out is to whether I should:

a) Max out to the permissible level of all my available buildings before I upgrade to next age or level, or
b) Immediately upgrade when the option is made available

Is there a general rule for this? Are there other things to consider?

a) Max out to the permissible level of all my available buildings before I upgrade to next age or level

To provide a general answer - this is a bad idea. This strategy requires more resources and more time, obviously. Not all upgrades are needed or required, and/or can be researched at a later point. You can even research when your town centre is advancing.

This will depend on your strategy. I mean, why upgrade pikemen if you never train them?

b) Immediately upgrade when the option is made available

This is often the case. However, I would consider other town centre functions before hitting the advance button. For example, do you have enough villagers? In later ages you can build multiple town centres so it isn't really a problem (i.e. you can advance off one town centre, and train/research through another).

Is there a general rule for this? Are there other things to consider?

To expand on "depending on your strategy" - every civilisation is different. Every map type is different (online/multiplayer). Every campaign level is different. However, to provide a general rule I would say advance as early as possible but don't sacrifice your town centre's other functions (mainly, training enough villagers). It is one thing to advance to the second age quickly, but if you arrive in the age with a weak economy you may never see the third age. This will mean you need to prioritise your research goals (maybe differentiate between "must have" and "nice to have" as well as "not required").

There are a number of things to consider when looking at ageing up.

What type of strategy are you playing?
Are you going for a rush (feudal/knight rush) or more of a long-game type strategy?
Can your economy sustain the growth from advancing to a new age?
What is your opponent doing?

If you're going to play a rush-style game, you need to advance quicker. If you're being rushed, you need to make sure you can get some defensive structures up. Researches and upgrades are expensive - you need to make sure your economy can produce enough resources to buy them.

Personally, I like to get to feudal age as soon as possible after reaching ~20 villagers, and then try to rush down the enemy, and only go to castle age once I have a well developed economy, or I need rams to take a town center.