Apart from going to learning centers, the best way to learn the local or the khaleeji dialect is to interact with the Qataris whenever possible. However, since the Qataris are a rare breed in their own country, the best alternative is to turn on the television and tune into some of the khaliji channels that air shows, cartoons etc for the children. Growing up in Qatar, I used to watch Iftah Ya Simsim, which is the Arabic version Sesame Street.

I also find watching khaleeji shows with subtitles very helpful. Netflix has got a few khaleeji shows. Try it out and good luck learning.