What online programming exercises would you suggest? Hopefully a site where you can submit code instead of submitting output.
It may not exactly be a submission-style system you are looking at, but one of the most popular programming exercise things out there is Project Euler where you can develop programs to tackle numerous mathematical problems.
On some of w3Schools.com's tutorials, you'll write XSL or javascript in your browser, and test it live...
If you are new to programming, I have seen this site used by many intro classes to teach basic programming. I am not exactly sure how difficult you want it to be, but you can take a look, I guess.
Great for learning python, but not for complete beginners or the faint of heart.
USACO
ACM ONLINE JUDGE
SPOJ
TopCoder - here you have to get Competition Arena , where you can find many training problems
- sorry , I'm new here , so I can't post links to those pages ... Use google to find them
Inner Workings isn't online but it does judge your code.
You have to pay for the tutorials (or drills as they call them) but they do offer free trials - you just have to check periodically to see if there any new ones.
Personally I'm not sure of their approach. I've only tried the free trial versions but in the areas I know I don't think they stretch me enough yet in the areas I'm not familiar with they don't give me enough information to get started.