I think small exercises would be the best way for me to learn programming. I am a beginner and I am willing to give plenty of time to this.
In my opinion, exercises should be something to write, and not long programs (10-30 lines).
What websites/articles/exercises can you recommend?
Project Euler has some sample problems that generally are solved with small programs in some cases.
Try Javabat (now coding bat I think). This is probably good for beginner exercises. Once you're getting comfortable with the exercises there (the recursion ones are particularly good), try Project Euler.
Python has a nifty puzzle way to learn the language: http://www.pythonchallenge.com/
You can look at Google's Python Class It is simple Python tutorial with some exercises. If you want to solve some puzzles then check Python Challenge. If you "dive" into those puzzles you will know many Python libraries (pickle, zip, Python Image Library)
There are a few suggestions over in the other SO post: Small but hard programming tasks (Complete apps as opposed to Euler)
From there, I suggested both Programming Praxis and Code Kata, and I'm personally working through exercises in SICP, which is a blast if you're interested in Scheme.
These may not be as newbie-friendly as some of the other options that pop up here, but they're all good resources.
The Facebook Careers website has a nice listing of medium-to-difficult programming puzzles, you can submit responses in any number of languages.
The problems sound simplistic at first glance but really have to do with things like graph traversal, bipartite graphs, string similarity scoring and edit distance, bin packing, etc.
I'm currently learning using the book "How to Think Like a Computer Scientist" which can be downloaded here: http://www.greenteapress.com/thinkapjava/. It has exercises after each chapter of explanation.
I think the Sphere Online Judge is still missing. It has hundreds of problems that can be solved in a multitude of languages.
Most of the problems are on the difficult side, though.
If you want beginner PHP exercises, try out phpexercises.com. It has 26 exercises, ranging from the very simplest "Hello World" scripts and moving on to control structures, forms, arrays and functions. You'll see the problem, then click a button to see the answer script after you've worked it out. There's also a link to the script output.