Algebraic geometry examples

The gluing along closed subscheme examples are a nice exercise for playing with Spec. That is, computing the spec of a pushout of affine schemes like
$$
(X leftarrow Z rightarrow Y)
$$
where one of the arrows is a closed immersion (or both are, for simplicity). It's a decent exercise to show that the Spec of such a pushout is what you expect pointwise.

Some examples that are worth trying as exercises are:

[1.] take a copy of $mathbb{A}^1$ for $X$, a pair of points for $Z$ and a single point for $Y$. You get a nodal singularity. Alternately, glue two copies of $mathbb{A}^1$ together at the origin.

[2.] $X = Spec k[x]$ as above, $Z = Spec k[x]/x^2$, $Y = k$, producing a cusp.

[3.] If $X = mathbb{A}^2$, $Z$ is an axis and the map $Z to Y$ is the usual 2-to-1 cover, then you get a pinch point.

[4.] You can make the map $Z to Y$ non-finite and produce non-noetherian affine schemes too. For example, $X = mathbb{A}^2$, $Z$ one of the axes (or any curve), and $Y$ a point.

You can later point out why you can't always do this for general (non-affine) schemes. That is, $X = mathbb{P}^2$, $Z$ an elliptic curve, and look at doing that sort of pinch point construction for various maps $Z to mathbb{P}^1$. This can lead to things like algebraic spaces if you are so inclined.

Another direction you can go with this sort of stuff is normality (ie, what is the geometric meaning of normality, all the examples you just computed with gluing are non-normal).