Following up on Sal Mangiafico's post, here are some more questions.
What are the two DVs?
Do you have two univariate questions or one multivariate question? (If you do not understand that question, see the Huberty & Morris article listed below. And if you are contemplating MANOVA, see the Huang article too.)
Are you interested in the interaction between your two factor variables, or just the main effects?
Were the groups formed via random allocation, or are they naturally occurring groups? If the latter, how unbalanced are they?
What statistical software do you use?
Article Multivariate Analysis Versus Multiple Univariate Analyses
Article MANOVA: A Procedure Whose Time Has Passed?
since you have two IVs with 4 groups each, it is a 2x4 factorial design I would say. The two-way ANOVA will allow you to examine the main effects of each IV as well as the interaction effect between the two IVs on the two DVs