From man bash
3.2.3 Lists of Commands
A list is a sequence of one or more pipelines separated by one of the operators ‘;’, ‘&’, ‘&&’, or ‘||’, and optionally terminated by one of ‘;’, ‘&’, or a newline.
Of these list operators, ‘&&’ and ‘||’ have equal precedence, followed by ‘;’ and ‘&’, which have equal precedence.
So, your example
echo this || echo that && echo other
could be read like
(this || that) && other