How to picture a first countable space?

Imagine a point and around it a sequence of ever smaller disks with center that point.

Whatever form you draw, if that point is in its interior there will be a disk small enough to be contained in that form, too.

If you need help to keep apart "first" and "second" countable, you could recall that the definition of 1st depends on what happens around 1 point.

I picture a (countable) sequence of shrinking neighborhoods around a point in $\mathbb{R}^2$. (Specifically, the balls $B_{1/n} (x)$.) If I draw any blob around this point, I just have to wait a bit and my neighborhoods will shrink enough so that they are eventually contained in it.

If something is first countable, then you can check if it is second countable. (Being second countable is stronger.)

And for being second countable, the picture is that you have countable collection of open sets that get fine enough so that any open set can be described as a union of some of them. For this I again picture $\mathbb{R}^2$, and fuzzily the collection of $1/n$ balls around the rational points, and maybe imagine taking some wiggly open set and filling it out with this little balls.