Im not sure if there is any application of set notation and set theory, however set notation is important when you start learning about the domains and ranges of functions.
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operations of sets in algebra
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It isn't; the algebra 1 and 2 that you get taught in middle or high school is elementary algebra. When (if you want to) you get into more advanced algebra, you can learn linear algebra (matrix algebra) and abstract algebra (which involves sets, operations on sets, groups, and many more concepts), and probably several more types of algebra I've never heard of.
The basic operations on sets are union, intersection, complement.
classification of sets