ANOVA 'factor level comparison' methods

We had an exercise in statistics class where we did a small ANOVA analysis. We were comparing how the two levels of a factor influenced the response under different conditions (combinations of other factors). My analysis ended up as a tree where a leaf corresponded to each condition and I could see were/how my factor of interest affected the response. It was quite tedious making sense of the R outputs even in such a small case (and the 'tree' was in my mind).

What techniques are usually used to make a 'factor-level comparison' like this in a more general case with lots of factors and levels? (Intuitive graphs and conclusions) Can it be generalized to ANCOVA?

What you are asking about is called "interactions", which are frequently visualized and hopefully understood with so-called "interaction plots": you plot the relationship between one independent variable and the fitted observations separately for each level of a factor. If there is no significant interaction, the lines will be parallel - if there is, they will not. Look at ?interaction.plot.