My book writes "tha urrainn dhomh". Why did you correct it?
Because it's wrong. It's definitely "is urrainn...", no "tha" there at all.
Also I've been taught expressions like "Nach mi a tha gòrach!" so I guessed it would be "nach a' Ghàidhlig a tha brèagha!". Why did you add "eil" after nach?
Well, because your sentence just didn't make sense to me
To use a similar structure like "nach mi tha gòrach" you could say "nach brèagha a tha a' Ghàidhlig". It's different anyway with nouns and personal pronouns, it's not completely the same pattern.
Thanks for your explanations about the stem/noun system. It seems quite clear. So "I want to buy apples" is litterally "I want the apples' buying" in Gaelic?
You can't translate it exactly literally, but as I wrote above it is something like "Is me at wanting the apples to buying". It does not make sense in English, I know, but you just have to get used to the Gaelic structures... don't do too much at once, step by step!
Another question now, is when do I use the verbal noun and when the stem? I think I need some explanation on the use and formation of tenses because my book won't explain them. Could you give me a table of Gaelic tenses and their formation (regular verbs only).
TAIC is really a great site, but again I'd suggest to go step by step, and *not* to start looking at the whole table of tenses! The most important thing (with every language) is to look at one structure and then practice it until you are really certain about it, then take the next one and practice again etc.
Listening and speaking is very important because it makes you remember patterns and structures and idiomatic expressions much better.
So, instead of printing out random lessons of TAIC, you should really go step by step, one lesson at a time, listen to all the stuff, repeat it, do all the exercises, etc.
A useful dictionary for a beginner would be the "Teach yourself Gaelic" dictionary by Iain Taylor and Boyd Robertson - do not confuse it with the learners' textbook of the same title. It has both directions, Gaelic -> English and English > Gaelic and covers all the vocabulary (with grammatical info) you need for a start.
And yes, there are certain patterns how to form the verbal nouns from verb stems, but for a start it's best to learn them with each verb and to practice them a lot
And yes, TAIC is done by a native speaker. Another good site for the sounds of Gaelic is http://www.akerbeltz.org
which is done by a non-native speaker, but he has better Gaelic than lots of native speakers