The two continuities

As you may know, there are essentially two versions of the Fullmetal Alchemist anime – the 2003 version, called simply “Fullmetal Alchemist”, and the 2009 version, called “Fullmetal Alchemist: Brotherhood”. The 2003 version deviates from the manga part-way through, while the 2009 version sticks to the manga all the way to the end. For more details, see What’s the difference between the FMA and FMA Brotherhood series?.

These two versions are unrelated to each other in the sense that it’s not the case that one is a sequel to the other, or anything like that. They’re just two separate continuities that happen to start in the same place.

This being the case, it’s up to you which one you want to watch first. Some people believe you should watch the 2003 version and then the 2009 version; some people believe the opposite; and yet others believe that you should only watch one and not the other. All these positions have their merits and demerits; perhaps the most typical order is 2003 followed by 2009, if only because that’s the order in which they were made.

Within each continuity there are multiple entries. There is a specific order in which these entries within a single continuity should be watched.
Within the 2003 continuity

You should watch in the following order:

Fullmetal Alchemist (the 51-episode 2003 TV series)
Fullmetal Alchemist: The Conqueror of Shamballa (a 2005 movie)

The Conqueror of Shamballa is a sequel to the 2003 TV series, so it makes no sense to watch it at any time before finishing the show.

There is also something in the 2003 continuity called Fullmetal Alchemist: Premium Collection. I have never managed to locate a copy of this, but it probably makes the most sense to watch it only after you’ve completed the 2003 TV series.
Within the 2009 continuity

You should watch in the following order:

Fullmetal Alchemist: Brotherhood (the 64-episode 2009 TV series)
Fullmetal Alchemist: The Sacred Star of Milos (a 2011 movie)

Unlike The Conqueror of Shamballa, The Sacred Star of Milos isn’t a sequel – rather, it’s a side-story taking place somewhere around episode 20 of Brotherhood. You could, in principle, watch it at any time after episode 24 of Brotherhood, but I would recommend leaving it until the end of the series anyway.

There are also four half-episode-length specials that accompany Brotherhood. You should watch these after you have finished Brotherhood.