At what point does the 2003 series and Manga/Brotherhood series start to diverge? I just finished watching Fullmetal Alchemist 2003 series. I would like to read/watch the manga/Brotherhood anime now. Does anyone know at what chapter the story of FMA2003 and the story of the manga/Brotherhood anime go their separate ways?
tl;dr: there is no exact point where the stories go &quot;different ways&quot;. Depending on how strict you are with the plot divergence, there are several candidates.
Minor differences
There are differences even at the beginning. For instance:

 In the Lior arc, we see an failed &quot;resurrection&quot; of Rose's boyfriend in 2003 that was created from some birds, whereas this never shows up in the manga. Al never even attempts the state alchemist qualifying exam in the manga, and the train and Shou Tucker incidents happen after Ed becomes a state alchemist. Roy finds the brothers because of reports of there having been skilled alchemists, not because of a letter.

There are also slight differences in character details (e.g. being a minority, Rose is darker in 2003). 2003 also has some  blatantly non-manga content at the beginning (e.g. episode 4).
When do major differences start to appear?
Looking at the Wikipedia chapter and 2003 episode, we have several candidates:

Ep. 7 (chapter 6, or ep. 5 of Brotherhood)


 In the manga continuity, it is specifically stated that Shou Tucker is killed. He is arrested, not killed in the original anime, and he shows up later.


Ep. 11:


 The Tringhams appear later in the anime so they can't just be regarded as &quot;filler&quot; material. But they're not in the manga continuity.


Ep. 14, 15 (thus, ch. 8/vol. 2, or ep. 4-5 of FMA:B)


 Grand is killed very early on in the manga, and only appears in Ishval flashbacks. Moreover the details about Ishval (which isn't discussed until chapter 58) and Scar are different.


Episode 21 (thus, ch. 11, or ep. 7 of FMA:B)


 Scar makes no such appearance in the manga, and the role of his brother is different. The events at Laboratory 5 are different.


Ep. 25, where the 2003 plot completely diverges (so ep. 10 of FMA:B or ch. 15).


 The details of Hughes' death are different. The homunculi are different in the manga continuity.

FMA:B and the manga plots seem unrecognisable from a 2003 perspective by around ep. 12-14 or vol. 8. However, important differences will have come up before that even though most of the events might be recognisable to someone who saw 2003. There's no single point before which everything is identical and everything afterwards is not. Moreover, some details in later chapters (e.g. the Elrics' childhood, in ep. 3 vs. part of vol. 5) are covered relatively early on in 2003, so there's going to be some duplication unless you specifically avoid those chapters.
Recommendations
If you want to be aware of all small deviations, starting from the first chapter/episode of the respective adaptation is best. If small differences aren't much issue and you want to reduce the amount you read/watch, vol. 2, ep. 4 (FMA:B), or ep. 10 (2003) seems like a decent points, although opinions will vary. Keep in mind that later plot will still be similar or even identical. If you skip stuff, it may be useful to read summaries and make sure you know the chronology of the adaptation in question.
Incidentally, The Sacred Star of Milos should not require much manga knowledge. I did not watch Conqueror of Shamballa closely, but my impression of it was that it required knowledge of the 2003 ending (ep. 48 onwards).