The Fubini type polynomials have many application not only especially in combinatorial analysis, but also other branches of mathematics, in engineering and related areas. Therefore, by using the $p$-adic integrals method and functional equation of the generating functions for Fubini type polynomials and numbers, we derive various different new identities, relations and formulas including well-known numbers and polynomials such as the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers of the second kind, the $\lambda$-array polynomials and the Lah numbers.