I did a post Just Starting Out, Which games should I make? which should be useful you now. It's a list of projects in escalating difficulty that I suggest beginners start with.

To add to that, there is also this article on GDNet http://www.gamedev.net/page/resources/_/technical/game-programming/your-first-step-to-game-development-starts-here-r2976

As both the links describe, but is not really explicitly pointed out, it is far easier to start with text games instead of graphics games.

One of the first games aspiring programmers can build is "guess the number". Pick a random number, allow a certain number of tries, and go from there. It relies only on simple text input and text output.

A popular game that is actually good for quite a lot of research is Tic Tac Toe. The first pass of text game, you can just scroll the old board up a few rows if you want. It is popular in college as a good tutorial on the minimax problem, which is a popular tool in game AI. Of course the game is solved already and it is easy to take the ideal move, but building an AI that calculates the ideal move is a good programming exercise.

Connect Four can be done with text, and is similarly good for simple AI research.

Hangman can be fun for humans and as a technical challenge for beginners, but as a source of study doesn't offer much.

Text based adventures are pretty easy using modern tools. In one of my articles I wrote several variations of a simple text adventure in a couple hours. Once you understand the science side behind it (state machines) you can make all kinds of text adventures. Advancing the plot is a matter of exercising a state machine until you get to a final state.