There is no one algorithm. Algorithms are defined programming steps (in this case) that perform a process. This is a very general definition, but so is the term. To go from the general to the specific, one has to choose a process first. Lets us say generating a log table to some unusual base like 6. Through refinement of the code, theprocess is made more efficient. The series if programming steps would then be the algorithm for generating base 6 log tables.
There are probably over a thousand different types and flavors of computer languages. An algorithm is just a word for a bit of that code or a whole program.
The Knuth book looks good, try and find it used. Goodwill and other thrift stores will sometimes have older textbooks which can be of use.
Of course a library is cheaper. That set used is about $60
You probably won't find Knuth's book at a thrift store (unless you live in silicon valley or are just extremely lucky). "The art of computer programming" is basically the equivalent of the Feynmann lectures on physics -- they are the basics of the field written by its most prolific practitioner. Any materials you can get from him or his website will be extremely valuable to you as a programmer.
I would suggest buying the book directly from his publisher / website. It is worth every penny if you can get your hands on it.
There are many types of programming, just as there are many types of cooking. Much of what you learn in one type of programming can be transferred to another type. There is "microcontroller programming," which is more about controlling your world, and is closer to electronics. There is "computer programming," which is more about making the computer do what you want. And computer programming can be broken down into "lower level," where you decide you want a window with so many buttons, and this text, and "higher level" such as scripts. (Note that I made up these names to describe general areas of programming. There is overlap between all the areas.)
Deciding where you want to start will affect where and what you look for.