In the given answers many possible solutions or options have been quoted.
Before adding a few I want to point at an important point:
In the 20-30 Arabic speaking countries there are at least 50 different slangs and dialects which are far more in use than the so called (modern?) standard Arabic or fasih language اللغة العربية الفصيحة (which is the -written- language of the Quran and appears in literature, in the TV-News and Newspapers and printed media in first place) which has reached the usage level of Latin in Europe. So basically the (sound or correctness of the) pronunciation of what people consider Arabic may differ from a Country / Region / Location to an other.
Therefore you should first consider defining or thinking of what exactly you want to do? Learn a dialect? Learn fasih language and for what purpose you want to use it. Basically nobody use fasih language in everyday life nor in colloquial language, but you might be confronted to it in occasions (such as display panels and as mentioned before in some media). And usually people speaking these "Arabic" dialects will more or less pretend that their dialect is close(r) to the fasih language.
So you must make this decision first.
One important way to learn the correct pronunciation of the Arabic letters is mostly -only- taught via oral transmission by Quran-memorizers, as before one should first learn the correct pronunciation of letters before learning how to correctly recite a Quran verse this usually is called Articulation of sound مخارج الحروف which are said to be fourteen, sixteen or seventeen (3 different opinions).
This is usually a matter of tajweed. And I don't know a better way to learn the correct pronunciation as good teachers of Tajweed are very severe and of course you need to practice in your spare time. (see also this link)