That assumes 'start' means 'start reading' though! You're adding information that's not in the original sentence. Check it out:
It has a range of general meanings, including creating something - so the sentence could easily be about starting that magazine you've always thought about. Natural translations are good but Duo's picky about accuracy as well, and adding 'read' in there changes the meaning and generality of the sentence
If it means "start reading that magazine" (as opposed to establish / create / found that magazine), I feel "I'm going to start on that magazine" should be accepted. http://www.macmillandictionary.com/dictionary/british/start-on Reported 16th June 2014.
They are pretty much synonyms. It is like asking what is the difference between start, begin, initiate, and commence. I'm sure there are small nuances that differentiate them, just like you would never say "Dinner will initiate at six" but I couldn't tell you what they are. You could look them up on SpanishDict.com and memorize some examples, but the nuances of words like these kind of grow on you as you read, watch movies, and speak with natives. Even if you use one of them in an odd way, folks will understand you.
"I am going to do something" is also, grammatically, present tense. Like the "going to" construction in English, the Spanish "futuro perifrástico" expresses a near future, and it's formed with a present tense form of ir followed by a and the full verb. "Ir a hacer algo" - "to be going to do something".
Iré would be "I will go" in English. This doesn't fit in this sentence.
If you have talked about a specific type of magazine before, saying "that magazine" is appropriate.
- "Remember how I was talking about how there isn't a weekly magazine about the history and development of cotton candy?"
- "I'm going to start that magazine. It's going to be a huge success!"