Being an aspiring Apple developer, I want to get the opinions of the community if it is better to learn C first before moving into Objective-C and ultimately the Cocoa Framework?
My gut says learn C, which will give me a good foundation.
I would learn C first. I learned C (and did a lot in C) before moving to Obj-C. I have many colleagues who never were real C programmers, they started with Obj-C and learned only as much C as necessary.
Every now and then I see how they solve a problem entirely in Obj-C, sometimes resulting in a very clumsy solutions. Usually I then replace some Obj-C code with pure C code (after all you can mix them as much as you like, the content of an Obj-C method can be entirely, pure C code). Without any intention to insult any Obj-C programmer, there are solutions that are very elegant in Obj-C, these are solutions that just work (and look) a lot better thanks to objects (OOP programming can make complex programs much more lovely than functional programming; polymorphism for example is a brilliant feature)... and I really like Obj-C (much more than C++! I hate the C++ syntax and some language features are plain overkill and lead to bad development patterns IMHO); however, when I sometimes re-write Obj-C code of my colleagues (and I really only do so, if I think this is absolutely necessary), the resulting code is usually 50% smaller, needs only 25% of the memory it used before and is about 400% faster at runtime.
What I'm trying to say here: Every language has its pros and cons. C has pros and cons and so does Obj-C. However, the really great feature of Obj-C (that's why I even like it more than Java) is that you can jump to plain C at will and back again. Why this is such a great feature? Because just like Obj-C fixes many of the cons of pure C, pure C can fix some of the cons of Obj-C. If you mix them together you'll receive a very powerful team.
If you only learn Obj-C and have no idea of C or only know the very basics of it and never tried how elegantly it can solve some common problems, you actually learned only half of Obj-C. C is a fundamental part of Obj-C. The ability to use C at any time and everywhere is a fundamental feature of it.
A typical example was some code we used that had to encode data in base64, but we could not use an external library for that (no OpenSSL lib). We used a base64 encoder, entirely written using Cocoa classes. It was working okay, but when we made it encode 200 MB of binary data, it took an eternity and the memory overhead was unacceptable. I replaced it with a tiny, ultra compact base64 encoder written entirely as one C function (I copied the function body into the method body, method took NSData as input and returned NSString as output, however inside the function everything was C). The C encoder was so much more compact, it beat the pure Cocoa encoder by the factor 8 in speed and the memory overhead was also much less. Encoding/Decoding data, playing around with bits and similar low level tasks are just the strong points of C.
Another example was some UI code that drew a lot of graphs. For storing the data necessary to paint the graphs, we used NSArray's. Actually NSMutableArray's, since the graph was animated. Result: Very slow graph animation. We replaced all NSArray's with normal C arrays, objects with structs (after all graph coordinate information is nothing you must have in objects), enumerator access with simple for loops and started moving data between the arrays with memcopy instead of taking data from one array to the other one, index for index. The result: A speed up by the factor 4. The graph animated smoothly, even on older PPC systems.
The weakness of C is that every more complex program gets ugly in the long run. Keeping C applications readable, extensible and manageable demands a lot of discipline of a programmer. Many projects fail because this discipline is missing. Obj-C makes it easy for you to structure your application using classes, inheritance, protocols and so on. That said, I would not use pure C functionality across the borders of a method unless necessary. I prefer to keep all code in an Objective-C app within the method of an object; everything else defeats the purpose of an OO application. However within the method I sometimes use pure C exclusively.