Are $X$ and $X+Y$ independent, if $X$ and $Y$ are independent? As asked in the title? Does the independence of two random variables $X$ and $Y$ imply the independence of $X$ and $X+Y$? If so, what's the easiest way to prove that?

No: If $X=\pm1$ with equal probability and independently $Y=\pm2$ also with equal probability, and you tell me the value of $X+Y$, then I can tell you the value of $X$ (and of $Y$).   
So in this case $X$ and $X+Y$ are not independent.  
And in general they usually will not be, as $\text{cov}(X,X+Y)=\text{var}(X)+\text{cov}(Y,X)$. So if $X$ and $Y$ have finite positive variances, then their being independent makes $\text{cov}(Y,X)=0$ and so $\text{cov}(X,X+Y)=\text{var}(X)>0$, meaning $X$ and $X+Y$ cannot be independent.
