You don't need both, and you shouldn't watch them back-to-back — they tell different stories from the same starting point.
For a first-timer: watch Brotherhood. It's the definitive version of the story. The only real argument for 2003-first is that Brotherhood rushes the early chapters (it assumes you've seen 2003), so some people do 2003 episodes 1–25 → Brotherhood from the start. That's a connoisseur's route, not a requirement.
If you love Brotherhood and want a different, moodier take on the same world afterwards, 2003 is absolutely worth your time as a second journey.
I was around for the original 2003 run back when it was still airing weekly, so let me give you the actual history instead of just "watch Brotherhood."
Hiromu Arakawa's manga started in 2001 and was still ongoing when Bones started the 2003 anime. Production caught up to the unfinished source material partway through, so the back half of 2003 becomes its own original story with its own ending — different homunculi origins, a different final-villain framing, a noticeably darker, more European-gothic tone than where the manga eventually went. It's not a bad show, it's genuinely well-regarded on its own, it's just telling a different story past a certain point, sharing only the setup and cast with what came later.
Brotherhood (2009) started once the manga was close enough to finished that Bones could adapt it front to back without running out of material, and it wraps up the same year the manga did. It follows Arakawa's actual plan for the story, full stop, and it's the version she considers canon.
You don't need to watch 2003 first, nothing in Brotherhood assumes you have. If anything, watching 2003 first primes you with a version of the ending it's going to directly contradict later. My honest take, and I'll get flak for it from other old-timers: watch Brotherhood as your main experience, then go back to 2003 afterward as a "what if" if you liked the cast enough to want an alternate take. Going in reverse order just makes 2003's back half feel like a downgrade instead of the different thing it actually is.
