If you want to learn Italian well, you need to read, and read a lot.
I write about reading here and elsewhere. Reading is probably the single most effective way to increase your Italian vocabulary. It is also the cheapest, which is why the purveyors of fine courses don't mention it much:)
Reading not only helps you learn high frequency words but also the lower frequency words that are crucial for good comprenehsion.
If is fashionable to recommend frequency lists such as the most common Italian words, as a way to hack or speed up language learning, and frequency lists for some languages are included on Surface Languages for that reason.
Personally, I consider frequency lists to be more helpful with the spoken rather than the written language.
If you read, you will naturally absorb the high frequency words as you progress. I haven't analysed this page, but the high frequency words such as 'the, this, and' occur again and again, and this will be identical in your Italian reading.
In my experience, the quickest way to increase your passive vocabulary is through reading, and I read in two ways. Firstly, I read for pleasure and secondly I read to improve my Italian.
When I'm reading for pleasure, I do not worry about understanding every word and the subtlety of the language. I want to be able to read my book, enjoy the flow of the language and the story or text that I'm reading. I will only look a word up where it stops me understanding the gist of a paragraph.
Reading for pleasure must improve my Italian, but I'm not actively attempting to retain new words and analyse grammatical patterns that may be new or unfamiliar to me.
I'd like to suggest a simple way to read and increase your vocabulary. Obviously, you can ... just read, and this is great. Find a book at your level and read it - using a dictionary where necessary.
Reading is a passive form of learning, and an easy way to make it active and retain more vocabulary, is to read a page of Italian, and then write a short summary.
I do it like this.
1. I read a page of Italian. I'm reading Le avventure di Alice nel paese delle meraviglie 'Alice in Wonderland'.
2. Try and write a summary in two or three lines.
3. Fail to do this.
4. Write down the ten most crucial words needed to write a summary. This keeps the process manegeable. My list, for example, for the first pages included tana 'lair', scaffale 'bookshelf', buio pesto 'pitch darkness' and scappare 'flee'.
5. Learn these words.
6. Write a summary in two or three lines. This is easy and effective.
A graded reader is a book written with text and vocabulary appropriate to learners at a specific level.
Graded readers are very useful for when you are starting out, a good Italian graded reader for beginner/low intermediates is Italian Short Stories For Beginners.
The trap to avoid is staying on graded readers for too long. It is important to make the leap to materials written for native Italian speakers as soon as possible, and so avoid remaining at the same level indefinitely.
Graded readers are great, but will only use a vocabulary limited to a few thousand words. It is natural that by sticking to these, you will miss out on the full range of expression and natural use of language that you will encounter using native materials.
You are spoilt for choice! There is a huge amount of Italian online. Try typing in 'libri italiani' for some suggestions.
If you want to read free books, try Project Gutenberg . It has a reasonable selection, although as these are books out of the copyright period, both the stories and language tends towards the archaic.
I prefer more recent novels, and nowadays read most of my Italian books such as Morte a Firenze on my Kindle.
Learning the Italian language: conversational fluency
500 Italian Sentences covering essential vocabulary.