Recently, I am learning the production of soluble and insoluble salts. My friend and I have done this experiment at the school lab. 

We wanted to taste them to see whether they are salty are not. The teacher luckily stopped us from doing that.

So without tasting them, I would really like to know whether all salts are salty.

No. There are sweet, bitter, and various other salts. (Likely, there are tasteless salts too). Pure salty taste is as far as I know exclusive for table salt, though I wouldn't bet on it.

Lead and Beryllium salts are said to be sweet, though toxic. Epsom salt, $\ce{MgSO4}$, is bitter. $\ce{CuSO4}$ has an incomprehensible, persistent metallic taste. (Based on personal experience. Copper salts are slightly toxic, but not extremely, so I survived with no consequences.)

Salts with hydrolysing cation (various alums) are acidic in addition to other notes.
