The computer craft wiki was already mentioned, but here are some more links.
http://www.lua.org/pil/1.1.html
This is a nice tutorial. Though, I never used it to learn programming I still use the parts about the more complex pieces.
It does however assume you installed lua instead of using computer craft. As such, certain things may not work (dofile is missing in computercraft for example). The differences are however minor.
http://www.computercraft.info/forums2/
A link to the computercrafts forum is of course not allowed to be missing. It has an entire section for tutorials, some directly programming related other just about apis and other about tools you can use. You can also get some programs here.
https://stackoverflow.com
https://stackoverflow.com
A site where you can ask questions about programming. It has both a lua tag as well as a computercraft tag. Note though that it is VERY much advised to search first before asking as there is a VERY good chance your question(s) where already asked.
Also, maybe give computercraft edu a look. It allows you to both write programs normally or write them using a drag and drop interface. This should reduce the amount of syntax errors you get.
Some other tips: Don't use normal computers to write your scripts. Instead use either an advanced computer or even better an external editor that you like. Both will give certain words that are part of the language a different color, making it very easy to spot if you typed something wrong. There are also some other editors for computercraft that have some extra features over the normal edit command.
If you want to use an external editor and are not on a server you can access the scripts you make while the game is running. I do recommend to make sure the computer/turtle is not doing anything though.
I believe you can find your scripts in
Code:
{your save}/computercraft/{computerId}/
If I'm wrong it should at least give you a direction on where too look. Note that a computer only gets a folder if it actually has something saved. Also, I believe an empty file will not count.
External editors that I used and though I may not use them currently they all served me well. Pick one you like
Notepad++
sublime
scite (This was when it came bundled with lua for windows and as such had been preconfigured specifically for lua.)
gedit with plugins (on linux at least, not sure how well it is on windows)
geany