There's no one-size-fits-all answer – it depends on what you want to achieve. The Arab world is diglossic, meaning two forms of the language exist side by side: in formal situations people use Modern Standard Arabic, in everyday life (on the street, in shops, within the family) the local dialect.
If you want to read literature or study the Islamic sciences, you need Modern Standard Arabic. If you have a personal connection to a particular country – say, because your family comes from there, you've married into a family there or you want to spend time there – it can make sense to start directly with the dialect.
Good to know: Modern Standard Arabic is understood in all Arab countries, even on the street. But with Standard Arabic alone, you won't understand everything that's said in dialect.
I myself learned Modern Standard Arabic first and Lebanese Arabic years later. The transition was surprisingly easy: Standard Arabic was my framework, the rest was mainly practice – speaking, listening and watching. If you want to master both in the end, in my view this path is easier than the other way around.