It depends what calculator you are talking about. The two most common I've seen/used are the TI-83+/84 (Just updates) as well as the TI-89.
On the TI-83/84, you are pretty much limited to TI-Basic or Assembly. TI-Basic is really easy to get going, but it will not be fast enough for a shoot-em-up. Assembly is your second route, there are a lot of good tutorials on the internet but it's hard nonetheless.
On the TI-89, it's pretty easy to get started, as long as you know C. There is an IDE you can install called TIGCC:
http://tigcc.ticalc.org/, which will output the binaries that you can run on your calculator. Rather than uploading the files to your calc constantly, TIGCC also has a feature that will automatically run the game in a calc emu, as long as you have it installed.
Hope that helps.