If we say fields $A$ and $B$ are isomorphic, does that just mean they are isomorphic as rings, or is there something else?
In a sense, yes, that is what it means. But not really. When we say two structures $S$ and $T$ of a certain type are isomorphic, we mean that there is a bijection $\varphi:S\rightarrow T$ which preserves the structure. So, for instance, if $\circ$ is a binary operation in the structure, then for $x,y\in S$, we have $\varphi(x\circ y)=\varphi(x)\circ \varphi(y)$.
It turns out that preserving the ring structure is enough to preserve the field structure; a field is just a commutative ring with inverses, so the property of being a field is preserved if the operations $+$ and $\times$ are preserved. Thus two fields are isomorphic if and only if they are isomorphic when considered as rings. But this is a contingent fact, and it's not really what we mean when we say that two fields are isomorphic.
I realise that this view verges on philosophy, and I wouldn't defend it to the death. I am just trying to give an idea of what mathematicians are thinking of when they say isomorphic.
They are just isomorphic as rings.
A ring isomorphism already preserves both operations of the field, and it's trivial to prove that a ring isomorphism "preserves inverses," so there's nothing else you could ask of an isomorphism between fields that isn't already there.