I have all three editions. The 1st one is 32-bit only, the 2nd covers both 32 and 64-bit and the 3rd is 64-bit only.
If you can, buy the 3rd edition (this is one of the few books worth its price) and then pickup the 1st edition for cheap/free from somewhere. Try for Used/Secondhand copies to save money.
There is a lot to learn from the book; all of them necessary for a beginner to connect the dots and get the full picture. So don't get stuck on any one chapter/problem but make sure to cover all the chapters in sequence skipping over unnecessary details in the first couple of passes. As an example you don't need to know the nitty-gritty of PIC/GOTs/PLTs/Assembly minutiae etc. in the beginning but just understand the concepts of Relocatable code and Executable vs. Shared Object files.
The authors have written up a summary [0] of the differences: in short, 3e reflects the changes in typical computer hardware over the last decade or two.
Note also that there are two versions of the third edition: the standard edition (colorfully striped shape on the cover) and the international edition (globe made of circuit boards on the cover) which is quite a bit cheaper. The contents of the chapters are the same, but the exercises at the ends of the chapters have been rewritten, by the publisher's lackeys, not by the original authors, for the international edition, very poorly by all accounts.
Do the labs too: these exercises are the most valuable. In particular, the binary bomb forces you to really understand the stack, heap, assembly instructions.
If you can, buy the 3rd edition (this is one of the few books worth its price) and then pickup the 1st edition for cheap/free from somewhere. Try for Used/Secondhand copies to save money.
There is a lot to learn from the book; all of them necessary for a beginner to connect the dots and get the full picture. So don't get stuck on any one chapter/problem but make sure to cover all the chapters in sequence skipping over unnecessary details in the first couple of passes. As an example you don't need to know the nitty-gritty of PIC/GOTs/PLTs/Assembly minutiae etc. in the beginning but just understand the concepts of Relocatable code and Executable vs. Shared Object files.