is there a way to glue two schemes together along a closed point (say we're working over a field)? Is it easier to glue in the category of algebraic spaces?

For this particular pushout, the geometric intuition is quite simple: given two algebraic varieties, one of which lives in $\mathbb A^m$, another in $\mathbb A^n$, combine them in two complementary hyperplanes in $\mathbb A^{m+n}$. Algebraically, this easily generalizes to an affine scheme $\mathrm{Spec}\\, R\_1\times R\_2/{\mathrm{relationship}}$ and then you glue everything together.

As correctly said above, general pushouts of schemes may not be schemes themselves.

is there a way to glue two schemes together along a closed point (say we're working over a field)? Is it easier to glue in the category of algebraic spaces?

For this particular pushout, the geometric intuition is quite simple: given two algebraic varieties, one of which lives in $\mathbb A^m$, another in $\mathbb A^n$, combine them in two complementary hyperplanes in $\mathbb A^{m+n}$. Algebraically, this easily generalizes to an affine scheme $\mathrm{Spec}(R_1\times R_2/\mathrm{relationship})$ and then you glue everything together.

As correctly said above, general pushouts of schemes may not be schemes themselves.