You might want to check out a tafsir while you read the Quran. They're extra notes to give context to the verses, so you know why they exist and what came about as a result. Ibn Kathir's one is pretty good, and is available here: http://www.qtafsir.com/index.php?option=com_content&task=view&id=3000&Itemid=731
You'll probably want to start learning to read Arabic so that you can recite the Quran in it's original language. Kids are taught with a book containing the alphabet and explanations for all the symbols. It's easy enough to learn if you have a book and some guidance, but takes effort to get your reading speed up. If you have a local mosque, ask them for help.
Lastly, get as much info about mainstream Islam as you can. Reddit's Islam subreddit has a great wiki outlining the basics at https://www.reddit.com/r/islam/wiki/index Don't get put off by it being from Reddit, it's worth reading. There's a Youtube series called iJihad that deals with general arguments against Islam, starting from the embed. We have a few threads like >>25688 that deal with other fallacies. There's also http://thecult.info/ if you come across any ahmadis, who are the Islamic equivalent to JWs and just as heretical. The more you learn about Islam the more you can recognise bad arguments, so learn as much as you can.