Standard Japanese language would be called 標準語(ひょうじゅんご), not 関東弁 (or any kind of ~弁).
Yes, if you learn to speak 標準語 well, you will be able to make yourself understood to the overwhelmingly vast majority of the Japanese populace, and you will (likewise) be able to understand them. The exceptions would be primarily older speakers in more remote areas with heavy dialects, who might be very difficult to understand even, on occasion, for native Japanese speakers (but obviously, more so for second-language learners).
Long story short, when you're just getting started, I would focus primarily (or entirely) on learning standard Japanese to the best of your ability. Once you have that base down, and if you ever find yourself living in Japan where a dialect (be it 関西弁 or 東北弁 or 北陸苑 or 九州弁 or what-have-you) are spoken, you can pick that up from friends and acquaintances as you feel necessary. There's really no reason to go out of your way to learn a dialect to any significant degree when you still haven't mastered 標準語. (And yes, there is a sort of old Tokyo dialect spoken primarily by older residents of the 下町 areas, but that's certainly not necessarily to learn to get by in Tokyo, any more than you'd need to master a heavy Brooklyn accent to get by in New York City, for example.)