I have a general question, but I'm going to give a very specific in-game example. So I played a game as an Anarch runner and I had Datasucker in play with multiple hosted virus counters. I also had a Parasite on a rezzed Data Raven (hosted power counters give the runner tags). My opponent's Data Raven had power counters on it. Parasite has the card text that when the ice it's on has strength 0 or less it is trashed.

I made a run, encountered the Data Raven and used the virus counters from Datasucker to drop the strength of Data Raven to zero which trashes it, due to Parasite's effect. My understanding is that the trashing resolves and the power counters on Data Raven are trashed along with it. My friend was trying to say that he got to use the hosted power counters to give me tags before his Data Raven got trashed… using logic similar to a MtG stack where my trash effect goes on the stack but his power counters go on 2nd and get to resolve before his ice is trashed.

How does timing of effects work? We couldn't find guidance in the rulebook.

There is no stack in A:NR. When several effects have the same trigger (see page 22 in the rulebook), the active player (in your example, it's you) resolves all of his, in the order of his choice (one at a time), then the other player resolves his, in the order of his choice (also one at a time). So you use your Datasucker markers first and Data Raven is trashed before the corp can use Data Raven's markers (which, on a side note, are best used during the corp's turn, because giving the runner tags is more powerful when it can be followed by punitive action).

One little detail that Khudzlin didn't address : in this scenario, you will receive a tag from first encountering Data Raven. The trigger for that effect is when you first encounter the ice, before you have a chance to do anything.

As for the other part, I think that the relevant rules are really on page 21 under Timing priority, not on p 22 under "Simultaneous Effects". I think the "Simultaneous Effects" text would apply in some future scenario where perhaps an Ice Breaker does something when initially encountering a piece of ice, and meanwhile Data Raven does something when it is encountered.

From the rulebook p21, Timing Priority,
Whenever there is an opportunity to trigger paid abilities, rez cards and/or score agendas…, the player who is currently taking his turn gets the first opportunity to act. He can trigger as many abilities, rez as many cards, and/or score as many agendas as he wishes in the order of his choosing. When he is finished, the other player gets the opportunity to act. When that player is finished, the first player gets the opportunity to act once again.

So before your opponent gets a chance to use the hosted counter, you have dropped the strength of the ice and Parasite has killed it.

Oh, and one other thing - your friend could have used the counter if he'd been a little faster.

Before you encounter Data Raven, you are approaching Data Raven. See the timing chart on p33. There is an opportunity for the Corp to use a paid ability (like that from the hosted counter on Data Raven) while you are approaching the ice. This is before the encounter.

So, the scenario would look like :

You approach the next piece of ice, and decide not to jack out;

Corp activates Data Raven to spend the counter and give you a tag

You encounter Data Raven and either take a tag or end the run

You activate Data Sucker to lower the strength, and Parasite kills the Data Raven.