Let $X$ be a non-singular (complex) variety and $Y \subset X$ be a (reduced) irreducible subvariety with only normal crossings singularity (locally, in the analytic topology, the singularity is defined by a product of the coordinates). Is there a simple strategy to "resolve" the singularities of $Y$ such that we obtain a proper birational map $\phi:\tilde{X} \to X$ which is an isomorphism on $X \backslash Y$ and $\phi^{-1}(Y)$ is a (reduced) simple normal crossings divisor in $X$ (under the Zariski topology)? Is it sufficient to blow up at the singular points? How many irreducible components will $\phi^{-1}(Y)$ have?

I am a beginner in singularity theory and the texts seem to suggest that normal crossings singularity is one of the simplest kinds of singularities, but I could not find a strategy on how to resolve it. Any reference on this topic will be most welcome.