Let $M$ be a $n$ dimensional manifolds, $f:M \rightarrow \mathbb {R}^n$ be a smooth map. Then, how can I calculate $\textrm{vol}(fM)$ ?
I'm thinking of calculating it using the area formula as shown below.
$$\int_{fM} dx = \int_M Jf(p) d\mu_g(p) .$$
However, the right-hand side does not make sense because the manifold M does not have a metric in the first place.
Am I doing stupid things?
I can answer a particular sub-case. You're right that you need a way of measuring volume on $M$ if you want this to work out. In particular, you need a metric (to generate the standard Riemannian volume form) or at least some top form if you want to integrate. Let $\mathrm{d}\mu$ be the standard Riemannian volume form on $\mathbb{R}^n$. Furthermore, assume that $f$ is an orientation-preserving diffeomorphism onto its image and disregard compactness issues (e.g. assume $M$ is compact). Then, $$\int_{f(M)}\mathrm{d}\mu = \int_M f^*\mathrm{d}\mu.$$ This is discussed in chapter 16 of Lee's smooth manifold book, so maybe you can piece together more details by reading that.