Cell motility represents one of the most fundamental functions in life such as organ development or tissue regeneration and diseases such as cancer. Here, we derived a computational model of cell motility that incorporates the most important mechanisms toward cell motility: cell protrusion, polarization, and retrograde flow. We first validate our model to explain the main phases of cell motility, i.e. symmetric cell spreading, cell polarization, and two important types of cell migration, confined and ameboid cell migration. Then, we use our model to investigate the effect of chemical and mechanical signals in cell migration, that is chemotaxis and durotaxis. More importantly, we analyze the competition between durotaxis and chemotaxis. We show that chemotaxis rules over durotaxis in most situations although durotaxis can diminish chemotaxis. Moreover, we show that inhibiting the effect of GTPases in actin polymerization at the cell front may allow durotaxis to take control over chemotaxis in soft substrates. Understanding how the main forces in cell motility cooperate, and how a precise manipulation of external cues may control directed cell migration is not only key for a fundamental comprehension of cell biology but also plays a key role in the future of tissue and biomedical engineering. To this end, we provide a freely-available platform to predict all phases and modes of cell motility analyzed in this work.