Often the best way to learn is by doing and dragon runes are no different, this method should hopefully have you able to read in a week or so depending on how often you practice. I used this alone however I would recommend checking out the lessons page here on Thuum.org if you haven't already. 

Step 1: First off, you're going to need a letter key. There is one here in the lessons page https://www.thuum.org/learn/grammar/alphabet.php and another on uesp wiki titled "dragon alphabet". Either keep this open in one tab, print it out or write it down so you can easily access it.

Step 2: Second you need something to read. The uesp wiki records all of Skyrim's word walls in runic, romanized and translated Dovahzul. Search for "Skyrim shouts" and click on the uesp page. (Or copy and paste this https://en.uesp.net/wiki/Skyrim:Dragon_Shouts) From here you can select the individual shouts to see their word walls.

Step 3: Enlarge the page so you can only see the runic part. Now read through it, checking your letter key whenever you don't recognize a letter (this will be all of them at first). I recommend writing it down as you go along to help you keep track. Once you finish you can scroll down and check your answers.

Step 4: Keep going! It's best to pace yourself so I recommend doing either one wall or one shout (usually three walls) each day. With each wall you will find yourself checking the key less often, by the time you've read them all you likely won't need it at all.

Step 5: Keep finding ways to practice such as writing, sikaav or boggle so that you don't forget what you've learned. Good luck and comment any questions you have and I'll do my best to answer.

This is absolutely incredible! This is an amazing way to learn runic Dovahzul! I hope I can use this to help me learn! If you come up with any more suggestions about learning it, then please continue posting.

@GronZeymah

Thank you. Do let me know if it helps. :)