First download NASM and then GCC. This is a good tutorial on how to compile them. I changed my global variables (or Path) so It will include directories of GCC and NASM so if I want to have the ASM file in another folder, I can still use NASM and GCC. And you can use any other compiler you want. (I just you NASM because it is the only thing I know). And this is all Assumes you use Windows. I can tell you how to set Assembler and GCC up on Ubuntu if you want to.
The first thing you need to know about assembly language is that it is specific both the the type of computer you are using, and to the operating system it is running. Furthermore, each assembler (the programs that convert assembly language programs into object code) has it's own variant syntax, and programs written for one assembler generally won't assemble using a different one.
If you are looking to write programs for Windows systems, and don't know what assembler you have or don't already have an assembler to use, then I would recommend following Sergent's advice. NASM is about the easiest PC assembler available, and several editors such as Notepad++ or SciTE have compatible assembly language modes for it.
If you are looking just to find out what assembly programming is like, and don't mind using a simulator, then you may find that SPIM is an easier solution; it simulates a MIPS processor, which is a type different from the one used by Windows machines, but is much easier to learn with. The simulator provides a good deal of additional support for debugging programs, and shows you exactly what is going on 'under the hood' as the program is running. The MIPS assembly language is vastly simpler than the x86 assembly, as well. The only hitch is, you don't get to see how a program would run on a real system, just on the simulator. Still, you could probably learn MIPS assembly language, then use what you've learned from that to learn x86 assembly for NASM, in less time than it would take to learn x86 assembly by itself.