The t-test can only compare one or two levels of a single factor. Thus, you are limited to: one-sample test (one level of one factor against 0), two-sample t-test (one level compared to another level of one factor, levels are from different subjects), and paired t-test (one level compared to another level of one factor, levels are from the same subjects.
For group*time (2 factors), you need an ANOVA model. For repeated-measures (e.g. time or condition), you need to use the flexible factorial model. Only the effect of the repeated-measure factor and interactions with the repeated-measure factor will be valid in this model.