Intel is what you'll most often encounter though so you do need to learn x86 and x86_64.
OpenSecurityTraining.info provides a number of courses that are valuable for this bridge to breaking software.
1. Life of Binaries - http://opensecuritytraining.info/LifeOfBinaries.html - This helps you go from understanding to software to understanding the system around the software and the context in which software runs. 2. Introductory Intel x86 - http://opensecuritytraining.info/IntroX86.html - Really boring/dry class on x86 instructions but gives you the introduction you need.
3. Introductory Intel x86-64 - http://opensecuritytraining.info/Intr oX86-64.html - Just slides this time, good to review and get a sense of the differences between 32bit and 64bit intel assembly.
Once you've gotten the basics its finally time to move onto learning the actual exploit development skills.
1. Introduction to Software Exploits - http://opensecuritytraining.info/Exploits1.html - In my opinion this is simply the best resource out there to learn the basics. It uses the book "The Shellcoder's Handbook" as its textbook and I completely recommend the book.
2. Hacking: The Art of Exploitation - This is the most often recommended book, its great and has a much better introduction than the Shellcoder's Handbook but if you can make it through the course above without probably you can probably skip this book as the two resources covers more content but this book is one of the best introductions available. 3. Corelan's Exploit Development Tutorial Series - https://www.corelan.be/index.php/2009/07/19/exploit-writing-... - It'll start in familiar territory but it'll get into some new stuff, overall a good series. 4. Exploitation in the Windows Environment - http://opensecuritytraining.info/Exploits2.html - You'll find some overlap with Corelan's tutorial series and this course so you might want to take this course and reference the tutorials as you go.
5. A Bug Hunter's Diary - Excellent book that covers some similar topics as the previous resources but spends a bit more time on actually finding vulnerabilities not just exploiting them and goes into more mitigations than the previous resources also, skip the stuff you already know. 6.
While learning all this exploit development stuff, there is another necessary skill to actually finding vulnerabilities: reverse engineering.
There are two books that I frequently recommend on the topic:
1. Reversing: Secrets of Reverse Engineering - this is the most popular recommendation and its a great resource to work through.
2. Practical Reverse Engineering - This is a new comer (2014) but I quite like it. It isn't as 'complete' as Reversing is but it covers a wider rage of topics that I find more useful.
3. (Bonus) Malware Analyst's Cookbook - Malware Analysis is probably the most RE heavy field you can be in so this is a solid book on the topic. Just because of its name I didn't give it a fair chance when I was reviewing books to recommend but I did review it recently and do want to give it a plug and it has a lot of practical information and labs to work on.
By this point you should have a reasonably solid foundation and a good understanding of exploitation. You will not be up to writing the latest browser 0day but you'll have the foundation necessary to understand (and learn from) modern sophisticated exploits so you can find and development them yourself. There are no resources to fill in the final gap but to go out do your research on a system and apply what you've learned to find some way to break them and development that weakness into an exploit.
To get experience, there are a few resources I can recommend:
1. Exploit-Exercises, I already mentioned Nebula, Protostar should be accessible to you once you've done the first Software Exploits course, and Fusion after the second one.
2. Over the Wire, I've already mentioned a couple of their servers, check out the rest of them.
3. Pwnable.kr - Challenges are at various levels use the harder ones to challenge yourself.
4. Capture-The-Flag competitions - every year several CTFs are run, sign up and play in them. What is nice about CTFs is that they are bite-sized challenges, still difficult, still involving modern techniques (the ones worth the most points atleast) but not tedious and they don't require a big time investment to find a weakness in. The focus of the challenge is on the exploit development rather than on finding vulnerabilities. 5. CVE lists - find software that interests you, find a known vulnerability and try to build your own exploit in it.
6. Real world software, go and break something of interest to you, learn how it works, find a vuln and exploit it.
You may need to learn a new language, or research some new techniques to handle some mitigations, but you should have the foundation necessary to figure out what you don't know and how to learn what you need.
...and with all this content I never even touched on breaking web applications, so I must atleast give mention to "The Web-Application Hackers Handbook" cover that book, practice against any of the many vulnerable meant to be hacked web-apps out there (Damn Vulnerable Web App, OWASP Mutillidae 2, HackThisSite, HellboundHackers, Enigma-Group, HackThis.co.uk, etc, etc)
Good Luck!