I'm beginning to transition from web development into programming and when I learn a new web development language, a common application for me to construct would be a blog or CMS.
What are some common applications to try out given any programming language?—aside from IO and "Hello World".
A common thing to do is to try solving some problems at Project Euler.
If you want to end up doing web applications though, it would probably be more constructive to do simple things like reading from a database, or dynamically generating a page using whatever parameters you want to use. Learn what you need to learn.
Simple Guess Number, has input, output, flow control and random inside.
Implementation of various algorithm, such as Pathfinding, Graph traversal, and so on. Implementation of various complex design patterns (such as the Observer pattern)
I'm not sure if networking goes under I/O for you, but otherwise that too.
Implementations could be things like, finding the shortest path from one address to another, implementing a maze solver (or generator), or writing your own chat client.
Heavily depends on the programming language you are trying to learn and on the operating system you are doing that.
System specific things I'd say. Read environment variables and do something upon a certain result. Then file operations like move files from A to B and compress them.
I always sit back and think about what time consuming operations I repeatedly have to do and then I try come up with a programmatic solution to save time.