Let $X$ and $Y$ be two continuous random variables with joint probability function given by $f_{X,Y}(x,y)=\frac{1}{x}\exp(-x-y/x),x,y>0$. Set $Z=Y$ if $X\geq1$ and $0$ otherwise. find $E(Z)$
So far, I have $$f_X(x)=\int_{y>0}f_{X,Y}(x,y)=\exp(-x)$$ and $Y$ is exponential(x) random variable. I was thinking of doing $EE(Z|X\geq1)$, but I don't know how to proceed after.
Your idea is a good start, but you do not need the conditional expectation. Just apply the linearity of expectation.
$\qquad\begin{align}\mathsf E(Z) &= \mathsf E(Y\cdot\mathbf 1_{X\geq 1}+0\cdot\mathbf 1_{X\lt 1})\\[1ex]&= \iint_{x\geq 1} y~f(x,y)\,\mathrm d (x, y)\\&= \int_1^\infty \dfrac{\mathrm e^{-x}}x\int_0^\infty y\,\mathrm e^{-y/x}\,\mathrm d y\,\mathrm d x\\&~~\vdots\end{align}$