How do aliens get stronger?

Over the course of a playthrough, what determines the strength and quantity of the alien units you fase on missions? For example, do the aliens grow stronger:

• continuously over time?
• at specific points in time (e.g. end of each month)?
• after each mission?
• as XCOM's squad ranks up?
• as XCOM's tech advances?
• as more regions are contacted?

A few implications of this are:

• If I skip a mission, am I letting the aliens grow stronger while I miss out on resources, or do both sides stagnate?
• Will pushing XCOM's tech forward only cause the aliens to advance as well?
• If wait to start a mission (e.g. a facility assault), do I risk it getting suddenly harder after a certain event (e.g. the end-of-month status report)?

Aliens grow stronger over time, and outside of a couple story specific aliens, aren't tied to mission or tech completion at all.

I believe the only story specific aliens are:

Codex, which appears after you skulljack a soldier, and the Avatar, which only appears when you skulljack a Codex and on the final mission.

Essentially every month you'll see some change in aliens which appear, which could include new aliens appearing or older ones disappearing, but could also could a frequency change in certain aliens (i.e. certain stronger aliens appear more often later into the game). Which aliens you'll see in which month depends on the difficulty, and sometimes you'll actually see changes happen partway through the month instead of the start of it.

So with all this in mind, there's no reason to hold back on research since it'll just put you further behind. You also shouldn't skip missions in attempt to slow down alien progress, as all you'll do is deny yourself resources and soldier XP.

Finally:

If wait to start a mission (e.g. a facility assault), do I risk it getting suddenly harder after a certain event (e.g. the end-of-month status report)?

Yes, missions which stick around will get harder over time. The aliens which appear on a mission is determined when you actually start the mission, not when it spawns.