In Fullmetal Alchemist: Brotherhood, the characters gain skill in alchemy through learning. So I'm wondering, is it possible to learn more than one alchemy? Because it seems like everyone is a specialized in particular alchemy. Also, if this possible, can they perform more than one alchemy at the same time?
Yes! Absolutely, characters can learn more than one kind of alchemy. It's just that, as in real life, people tend to specialize in something very specific and get very good at it.
But let's take a look at a few examples:
Then there's a few oddballs, but the rule can still apply to them:
Keep in mind that, similar to what @kaine said, transmutation circles come into play. Not just in knowledge, though; the person actually has to either have that transmutation circle with them or draw it on the spot, which is not feasible in combat. So, someone like Armstrong would not thrive at the same things as Basque Grand, because they both use different transmutation circles. This is clear with Mustang who broadens his use of alchemy after his "accident".
So from this, we can tell that it's possible for alchemists to learn more than one type of alchemy, but it's difficult for them to always apply it under pressure (for multiple reasons); given cases like Ed, who doesn't need transmutation circles, we can see a broader range of alchemy used.
Focusing on Edward, he does alchemy (at least) on metal, binds souls to armor, attempts human transmutation, repairs machines, changes carbon structure (in Greed), etc.
I believe that he does alchemy on what he feels useful. Everyone else is only limited by what circles they have already drawn (as they need transmutation circles to do alchemy) and their knowledge. In other words, they need to know the characteristics and how to draw the circles for what they are transmuting and what they are transmuting it from. For instance Mustang handles fire as that is what transmutation circle is on his gloves.
TLDR: No, they are not (usually) only limited to one type of transmutation but they have to have the circle and knowledge for any transmutation they are doing.