What is the difference between [verb + [I]koto], [verb + [I]no] and the -ite/-ide/-shite/-nde/-tte/-te ending of verbs?
From what I understand, [verb + [I]koto] and [verb + [I]no] are the equivalent of the English gerund and can be used interchangeably. I also understand how the -ite/-ide/-shite/-nde/-tte/-te forms are made and how they function, but I don't really see how these three things are different as words. For example, kaku-no, kaku-koto and kaite all mean "writing", right?
I did a quick search and found this, if this helps for the no vs koto
I don't have access to my Genki 1 or 2 textbooks (away from home for a while) but it does cover this quite well. I'll try to compare.
Kanji wo kaku no wa muzukashii desu
Kanji wo kaku koto ga muzukashii desu
Both mean It's difficult to write Kanji / Writing Kanji is difficult
Kaku no (wa) basically changes the current sentence to "The action of writing"
Kaku koto and no wa form
"Kaku koto by itself would be "(the action of) writing…"
This is talking about the action of writing, usually explain something like;
"Kono pen wo kattara kanji wo kaku no wa kantan ni narimasu"
"Kono pen wo kattara kanji wo kaku koto ga kantan ni narimasu"
If you buy this pen, writing Kanji will become easier.
Kyou wa, tegami wo kaite, okurimashita
Today I wrote a letter and sent it.
The te form is really just to connect a bunch of sentences together. Really all it is.
Kaiteiru literally means to be currently writing.
Ima ha tegami wo kateiru / kaiteimasu (I'm writing a letter now")
I think this is the Kaite form you are thinking of.
As for the te forms (ite/-ide/-shite/-nde/-tte/-te ) you seem to be breaking down the different way the te form appears, it would save you a lot if you learnt them word by word.
The best way to learn it is when trying to use the te form of a verb, use a dictionary to confirm if it's correct everytime. I use JED on smartphone, shows all the forms of the verbs and easily searchable, doesn't require to press enter button to search, requires no internet and it's free.
I hope I helped, feel free to ask more questions.
–- Update From New Post Merge ---
Also, I would like to try to read some really, really simple children's books (preferebly in hiragana or romaji). Does anyone have some links to places where I could find those? (If something like that exists at all.)
Hmm, it is pretty difficult to find online, I've tried searching, but maybe not thoroughly enough. Anyway, you can actually try pick up a manga to speed up hiragana and katakana reading, I did that at first with a manga that didn't use furigana which was pretty difficult at that time. You could even try One Piece purely for the purpose of speading up Kana reading and would be fun since it would be One Piece.
Shonen manga all have furigana. If you want access to some Japanese raws of One Piece you can access it online here , thanks to Sandman for the link.
One other great way to practice Kana reading would be News web easy https://www3.nhk.or.jp/news/easy/ , I would highly recommend you read literature that has furigana at least, plain old hiragana won't give any Kanji exposure.
News web easy offers latest news articles in which you can also find the original article if you wish which is from NHK News. I personally don't like reading there anymore since furigana is so distracting, I ditched furigana 2-4 months in and got used to going reading and searching words myself.
I did read a lot of religious publications on Jw.org (religious website). You can get the English translation of any publication, the Japanese version along with the furigana Japanese version. This is as top as you can get in translation and I personally used this and am currently using this. Only downsides is it is really complicated but I did speed up my reading and learnt a ton of words.
http://imgur.com/1JOY0kH Pure Japanese
I find it's really hard to find articles in all three formats online.