Script
The Persian alphabet usually scares new learners, but it really shouldn't. If you aren't fazed by having to remember thousands of new words in a new language, then you shouldn't have any trouble memorizing the mere 32 letters you'll need to read the script. You'll be able to read your first words within a couple of hours.
Of course, being able to decipher a word letter by letter is not the same as reading fast and fluently - that does require a lot of practice. At the beginning, you'll find it frustrating how slowly you are reading. There is no doubt that this makes learning an otherwise easy language quite a bit harder. If you are considering not learning to read and focusing on conversational skills instead, have a look at our thoughts on the matter.
The resources here have been specifically designed to teach you to read and/or write Persian.
Learn to Read Persian in 5 DaysShort little booklet with exercises
This little book really is short enough to easily read through it in 5 days. It's really very no-nonsense, focusing on essential explanations and then giving you a bunch of words to read. with the letters you already learned.
Nothing more is really required. You'll want to practice more once you are done with this book to become a fluent reader, but this is the core instruction you need.
Persian Script HackingComprehensive & Fun
I love this book. It teaches you a single letter, then has you practice it by reading and writing. Also fun: You'll be asked to find the letter in a block of text, which is, I am told, what children do as well when learning the alphabet. For reading, you'll be reading words such as burger or Martin Luther - so no need at this point to actually understand the Persian language.
How to Write in PersianA Workbook with lots of space to practice writing
This is a nice book that walks you through the individual letters, shows you their stroke order, and then gives you a lot of lined space to practice writing.
A very nice touch I think is that they show you each letter in a couple of different fonts, to give you some idea about the different varieties of shapes you may encounter in real life. But I just do love fonts, so I may be overestimated how impactful that will be on your learning.