Z-transform of an FIR filter
If you know (or better: understand) the basic $\mathcal{Z}$-transform relation

$$x[n-k]\Longleftrightarrow z^{-k}X(z)$$

then from the given time domain equation you immediately get

$$Y(z)=X(z)+2X(z)z^{-1}=X(z)(1+2z^{-1})$$

Since the transfer function $H(z)$ is the ratio of $Y(z)$ and $X(z)$, you get

$$H(z)=1+2z^{-1}$$