Use && operator,
cmd1 && cmd2 && cmd3
In shellscripting, && and || operators are modelled after optimized implementation of logical operators in C. && means AND operator, and || means OR. The key point is that in C, the second operand of these operators isn't evaluated if the result is already known from the first operand. E.g. "false && x" is false for any x, so there is no need to evaluate x; similarly for "true || x".
And in Unix, it is traditional to interpret commands' return values as "successful completion" truth values: 0 means true (success), nonzero means false (failure). So, when the first command in cmd1 && cmd2 returns "false" (nonzero exit status, which indicates failure), the whole command's status is known: failure. So overall interpretation of cmd1 && cmd2 shell command may be: "execute cmd1, AND THEN, if it didn't fail, cmd2". Which is what you basically want in your question.
Similarly with OR: cmd1 || cmd2 can be interpreted as "execute cmd1, OR IF it fails, cmd2".