Start with the beginner's FAQ...

http://www.gamedev.net/page/reference/faq.php/_/for-beginners-r1

Well a game engine isn't really going to help improve your "coding/programming" skills. It can help you make a game and make the more tedious processes easier and let you focus on the game making itself. Since I assume you want to break out of the land of "ti basic" and move into a more traditional language you can start there. Once you have a language picked out start learning the syntax and basics of it. If you can remake those games you mentions for the Ti 84, I would say you have a good grasp of the basics. Now you can start looking at the many many different game engines they have available that might use those languages. If you want a recommendation of a game engine that still allows you to use a traditional language you can go Unity it will do 2D and/or 3D with a pretty sweet UI and allows you to use a few different languages for scripting.

Hi,

There are many 2D frameworks, some of which are not game engines but libraries of basic low level coding that you can use for graphics, input, sound, etc.. For a beginner the language is very important. Most beginners should avoid C++ in my opinion for the first several months at least. You really need to research carefully what exactly is the complete workflow pipeline of a game engine. Some are stand alone pipelines while others require added frameworks, SDKs, libraries, software, and tool applications . It is very important for you to choose a game engine with a workflow pipeline that fits both your talents and your skills. Start with outside view/ single player games then add views and multi-player on later games. Try to avoid spaghetti coding and gradually modularize your coding, for example, with class files, maybe a bin folder, both development and user interfaces, and so forth. Modularized coding is much better for extending and debugging. Comment your coding lines well and document module outside, too, so you can return more easily later to work with it.

The reason why I mentioned all of this is because a good game engine is fairly efficient at allowing these things. Your long term goal should be aimed at version control software in about 1-2 years. You will have to prepare for the advancements that will come with research and careful software planning. Flowcharts and other diagrams are crucial early to establish a habit of purposeful development.

Here is a list of game engines...

http://en.wikipedia.org/wiki/List_of_game_engines

Clinton