How to read letters such as $\mathbb A$, $\mathbb B$, etc., or $\mathfrak A$, $\mathfrak B$, etc.?

If there's a reason you choose to call them all A's, there's presumably a reason why some of them are fraktur, some of them are black board, and some of them are calligraphic.

Then don't read the font. Read the meaning. For example, when I write

$g(v_1,v_2) = \langle \mathfrak{g}_1,\mathfrak{g}_2\rangle$ where $\mathfrak{g}_1,\mathfrak{g}_2\in \mathfrak{G}$ correspond to $v_1,v_2\in T_p\mathcal{G}$

I don't read

italic gee of vee one and vee two is equal to the angle bracket of frak gee one and frak gee 2 where frak gee 1 and frak gee 2 belong to frak big gee and correspond to vee 1 and vee 2 in tee-pee cal gee.

That's completely incomprehensible!

I read

The action of the metric gee on vectors vee 1 and vee 2 is set to be equal to the Killing form acting on gee 1 and gee 2, where gee 1 and gee 2 are elements of the Lie algebra gee corresponding to the tangent vectors vee 1 and vee 2 in the tangent space at point pee of the Lie group gee.

In other words, the usual reason one uses different fonts is to help distinguish between different classes of objects (and in some case isolate special objects like $\mathbb{R}$ for the real numbers). In that case, when you are reading the text, you should speak the name of the class, not the name of the font.

$\Bbb A$ is blackboard A; $\frak A$ is Gothic A.

If there is only one $A$ being used, though, we read it as A regardless to the font.

It's surely a cultural thing, but I've heard $\frak{A}$ referred to as "fraktur A" more frequently, although I do hear "gothic" occasionally. On a similar note, I'm increasingly hearing people actually say the names of TeX commands, even for symbols (not just fonts).

For instance, I have a professor who says "sim-eek" every time he write $\simeq$ (LaTeX command simeq.)