There isn't a list of "most frequently used" algorithms: there are too many possibilities.h
Remember, an algorithm is
Google:a process or set of rules to be followed in calculations or other problem-solving operations, especially by a computer.
So almost anything we do as developers uses an algorithm of some sort, sometimes explicitly, sometimes implicitly. Most of the time we don't even think about "algorithms" per ce, we just get on with the job of implementing the specification in the most appropriate way.
But if you want "basic exercises" to get you thinking about how to do things, then start by implementing the basics: lists, queues, stacks, hash tables; then start manipulating them: searching, adding, deleting, sorting.
Don't focus on algorithms so much, focus on what you can do with them when you create applications.
Implement some of the basic homework tasks: a calculator that supports brackets and operator precedence, music libraries, maybe a calendar program with birthday reminders.
The best way to learn is go on a course / get a good book and do every exercise!