What are examples of simple problems and applications that can be solved with AI techniques, for a beginner who is trying to make use of his basic programming skills into AI at the beginning level?
This is fairly boilerplate advice, but, since you're brand new to AI, I'd personally suggest writing a classical Tic-Tac-Toe AI, ideally using minimax.
I suggest this because minimax is fundamental to AI, and there are many webpages devoted to this subject, such as How to make your Tic Tac Toe game unbeatable by using the minimax algorithm and Tic Tac Toe: Understanding the Minimax Algorithm. (Google search for "Tic-tac-toe" and "minimax" will yield a plethora of other sites. I'd also recommend looking at this minimax page from Stanford: "Strategies and Tactics for Intelligent Search".)
I recommend this approach as a good basic primer. The real cutting-edge work is being done in Machine Learning and Neural Networks, and for that reason, it's probably more important than ever to have some basic grounding in classical AI before you start dipping your toe in that pond.
I will assume you talk about applied AI (in generalized/strong AI we have nothing yet to program :-).
You can look at any university course of introduction to AI and see its chapters and the program examples they use ( start by programming without any theory is not a way ).
By example, one common issue on this kind of courses is path finding, using for it algorithms as A* algorithm and applying them to games as Hanoi Towers. This kind of knowledge is a must for any activity in AI.
Standford link provided by @DukeZhou is a good example of one of theses courses, just I suggest start it from first chapter instead of go directly to minmax.
Later on, you can jump to more advanced concepts, as recognition/classification and its common approaches: k-nearest/k-means, decision networks, neural nets, ... .