I am trying to use beta maps for second-level GLM analysis.
However, I cannot find a way to save the beta maps from the first-level GLM function. Does anyone know how to save the beta map as nii file?

I also found that nilearn default setting of the 2nd level ‘effect_size’ map. Is ‘effect size’ map beta map?

Pure beta maps will come from individual conditions (that is, those that go into a contrast). The effect size here would be analogous to the “con_XXX” maps from SPM.

Yes. You have beta maps for your individual conditions, contrast maps for the difference between the beta maps, t-maps for the statistical significance of the contrasts, and z-maps for a normalized version of the t-maps. Outputting the z-score will get you directly to the last step, which will likely be useful for the next step in your analyses.