I am using deseq2 to find differential significant taxa (count data). The data is longitudinal and three groups of mice and I am using sth like

diagdds = phyloseq_to_deseq2(physeq.fecal, ~ condition + week_experiment +
                             team + condition:week_experiment)

since I have three groups, I wonder how DEseq2 is calculating the fold change and p-value !? There is neither any warning nor any error ! should I slice the data 3 different combination of each group ?!

The second question I have is about condition:week_experiment term. I wonder how I can set it up in a way that two groups have the same initial value at the beginning (same intercept).

I'd recommend that you partner with a statistician who can help set up a design that works for you and interpret results. There are in fact many different designs you could use which "work" in that they don't produce an error about rank-deficiency or linear dependency between columns, but that doesn't mean these are the right design. And furthermore, you will need to know how to properly interpret the estimated coefficients or build contrasts.

I think, I haven't explained myself well, let me try one more time. Bacterial abundance of three different groups across 10 weeks is measure. Therefore, I chose this formulation/design to answer what are the taxa( in RNA world genes) that are distinguishing these three groups over the time.

diagdds = phyloseq_to_deseq2(physeq.fecal, ~ condition + week_experiment +
                             team + condition:week_experiment)

 In this setup, I wonder how does the fold change is calculated when there are three conditions, which I think I should look into the results function and it depends on the contrast, 

res = results(diagdds, contrast=c("condition","A","B"), cooksCutoff = FALSE)

I think I got my answer now