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Pro Industry standard
C is the industry standard programming language, moreover, it is the most popular programming language to use. C is the language used for most Windows, UNIX and Mac operating systems.
Pro Understanding of computers
Learning C forces you to grapple with the low-level workings of your computer (memory management, pointers, etc.) in ways that the other languages abstract away. Without an understanding of these low-level aspects of computer programming you will be limited if you try to develop an application that needs to run in a memory or performance constrained environment.
Other languages like Python can obscure a lot of details, so your foundation may be weaker.
Pro Helps with learning other languages later
Pro Must have
Capability to program in C is greatly appreciated in developers, creates an image of competency, and many programmers will learn it at some point in their careers.
C is portable between most hardware. Generally a C compiler is made for any new architecture, and already exists for existing architectures.
C is portable between all operating systems (Windows, UNIX and Mac) and only needs a program to be recompiled to work. This allows anyone on any operating system to learn about the language and not be held back by intricacies of their operating system.
With this said, C's portability these days is not quite what it used to be. Much of said portability relies on the POSIX standard in particular, and as time passes, the compliance of a given system with that standard is becoming less certain; especially in the case of Linux. Most things will still be portable (or at least emulatable) between Windows, Linux, and FreeBSD for example; but you will at times need to make use of platform-specific support libraries for certain individual cases, as well.
Pro If you can't grok C you should not be a professional programmer
It sets an early bar that if you can't hurdle you might as well do something other than programming and not waste any more of your time.
Pro Teaches good practices
Writing in C will require you to understand how things are done. C implies using and understanding the fundamentals. Learning a higher-level language after is much easier.
Pro Low level of abstraction
While higher level languages languages like Java and Python provide possibilities to be "more expressive" per line of code, it's much more convenient to start with "less efficient" (get me right) language, in order to get initial concepts of how things behave at lower level.
Actually C is a good starting point moving to both higher and lower levels of abstraction, the good example here would be learning C before Assembler, as for general use the Assembler quite hard to understand due to low level of its abstraction (like getting the understanding on how loops work in C before trying to implement them on Assembler).
Con Requires memory management
Learning programming is already hard enough when you don't have to worry about memory leaks.
Con Long learning curve
While the language compliments knowledge of computer components very well, and gives a deeper understanding, it is also quite difficult to learn, and to use correctly, especially without aforementioned knowledge.
Con C will require you to learn concepts too advanced for most beginners
While other programmers will learn algorithms and structures and will do magic tricks and awesome applications, you will learn trash info that you should know maybe after 5-7 years experience in software development, not earlier. It's like going the first time as a seven year old kid to first school class, and your teacher tells you to learn you about Discrete Math, without basic math and how to do 2x2.
If you wish to be a really good programmer, C for sure will be in your portfolio, but not as a first language, and this programming language is used only for very hard and very limited tools which require a lot of professional skills from the programmer.
Con Heavily outdated programming concepts
C lacks a large majority of programming concepts that modern languages make use of today. The existing functionality of C makes use of outdated and deprecated methodologies which can be of great annoyance to the modern day programmer.
Con Completely lacks type safety
The C standard library is not type safe, and the language itself does not promote type safety built into the language, which leads to error-proneness of the language. If anything, I would recommend those interested in C to instead put their time in D, which actually includes a complete copy of the C standard library rewritten to be fully type safe.
Depending on the purpose this can be either a pro or a con. If the task is to learn how to program, low-level of C will impend learning important concepts. Furthermore, C is rather limited in ways of building abstractions.
"Low-levelness" of C can be a pro feature in learning system programming.
Con Does not support modules; header file annoyances
Header files are a poor man's implementation of modules. Modern programming languages make use of modules which eliminate the need for C includes and header files and the many issues caused by them, such as the complete lack of dependency checking. Header files often contain even more include statements that point to other header files which also point to even more which drastically increases compile time. Modules only have to be compiled once, and when importing those modules into your software project, you only have to pull in the module that you are using, which is often times already precompiled. This way, the compiler knows exactly what it needs before beginning to compile your project and can automatically compile the few dependencies it needs in advance rather than recursively compiling every header file it runs across as in C.
Con Lack of support for first class strings
C does not support the string type, nor does it support UTF-8 strings that modern languages are employing today. Instead of strings, C makes use of the *char type which is a pointer to a character array.
Con Languages is full of corner cases and undefined behaviors
Undefined behavior in a program can cause unexpected results, making it hard to debug. With UB, program behavior may vary wildly depending on optimization settings. There are many cases that invoke UB, such as signed overflow, invalid dereferences, large integer shifts, uninitialized variables, etc. These serve to frustrate novice programmers when they could be learning other concepts.
Con The need for C developers in the current market is very low, and trending downward
Older languages, like C, are no longer in their hay day. Even if you do learn it as your first language, you are only setting yourself up to need to learn another language in the long run. If you want a skill that you can not only learn from, but also potentially build a career on, C should not be your first choice.
Con Arrays are not first class objects
When an array is passed to a function, it is converted to a pointer, even though the prototype confusingly says it's an array. When this conversion happens, all array type information gets lost. C arrays also cannot not be resized, which means that even simple aggregates like a stack are complicated to implement. C arrays also cannot be bounds checked, because they don't know what the array bounds are.
Con C structs are very weak and outdated
C structs lack a lot of modern capabilities that are vital in programming languages of today, such as assigning member functions to structs to give them object-oriented capabilities, constructs, deconstructors and RAII. Great care must be used when using structs in C to prevent memory leaks and ridiculously slow structs.
Con Isn't truly portable or cross platform
The C programming language is not portable to other operating systems, and even different compilers, because the C language does not provide any reference cross platform libraries or compilers. Different platforms and compilers provide their own implementation of the C standard library which may not be compatible with the implementation in another compiler or platform. Without cross platform libraries and tools, one cannot state that C is portable. This is in stark contrast to modern programming languages that provide their own cross platform libraries and compilers, such as D, Go and Rust.
Con Compiles procedurally rather than intelligently
In the same manner that C recursively compiles header files ad infinitum without any sort of dependency checking, C source code is also compiled in the same manner. If you attempt to call a function before it is declared, the compiler will fail because the function was not compiled before it was caled.
Con Includes require obscene resources to compile
All the modern languages have resulted in ditching the ancient deprecated model of #include statements and have instead adopted the superior model of modules. When compiling software written in C, the programmer is forced to also compile X headers which contain Y headers which contain Z headers and so forth -- drastically increasing the number of lines that need to be compiled. In order to compile something as simple as "Hello, World", for example, 18K lines of code needs to be compiled. This can be very taxing on RAM and CPU resources, causing compile times to quickly absorb a large portion of the programming process.
Con Lacks support for source code split into multiple files
Modern programming languages are capable of compiling split source code files by concatenating them together efficiently at compile time before compiling them. However, C does not support this functionality, instead requiring the developer to resort to messing with header files and makefiles to get similar functionality.
Con Hard to maintain in large software projects due to lack of unit testing
All modern languages worth their salt implements unit testing built into the language so that programmers can implement testing coverage for their entire software project to prevent bugs. However, C does not support unit testing built into the language, which makes the already error-prone language even more error prone.