Assalam-o-alikum,
Dear Sister Judit,
Maybe you already know the details about Quranic verses, but still allow me to give you some information regarding a website which explains Quran. It helped me a lot too.
As you already know Quran is not like other books where one chapters discusses one matter and the next chapter discusses another matter in sequence. Quran was revealed to Prophet Mohammad (PBUH) in verses, these verses were assembled togather in a way that later verses were inserted according to the instructions of Allah. They were then divided into chapters according to His instructions. So one chapter may discuss several subjects, therefore it helps if someone explains them and provide some background information about those chapters and verses.
In the following website you can find a very good translation as well as explanation of Quran:
http://www.tafheem.net/main.html
On the left there is an Index of chapters and on the right there are contents.
If you want to know the background of a chapter then click "Intro" (which is at the top) and read. Some chapters are long so they are divided into pieces e.g. verse [22-41] etc. It is easy.
You can also use this as a quick reference for the background of a verse or its explanation at any time. Hope this will help you.
Good Luck Sister and welcome to TTI.
Wassalam.