If you can stomach it, Qur'an helps a lot. Listen to a good reciter like Mshary Rashid al-'Afasy - the slow pace of the recitation makes it easier to learn than listening to newsreaders talking at 100mph, and they take pains over correct enunciation. It also has the benefit of being able to have the Arabic text in front of you as you're listening, plus Qur'an is always written with taskeel/vowels.
When I was learning MSA at university, I used Elementary Modern Standard Arabic by Peter F. Abboud and Ernest M. McCarus (I got through vols. 1 & 2, though I have shamefully now forgotten almost all of it). It's quite an old book, and the typesetting and fonts are not very pleasing to the eye, but generally speaking I found it well thought out, easy to work through and very helpful.