If you are looking for good first examples, Mumford's Red Book and Eisenbud and Harris's 'Geometry of Schemes' have some good pictures and examples.

Its worth playing around with Spec(O_c), where O_c is the ring of integers in the extension of Q by the square root of c, and thinking about it as a scheme over Spec(Z). In particular, several somewhat mysterious number theory terms like 'ramified' and 'split' make geometric sense in this context.

Its also worth thinking about what the p-adics should look like as a scheme - a formal neighborhood of p in Spec(Z) (though to make this precise you need to know what formal schemes are).

Its also not a terrible idea to pick up a book on algebraic number theory and try to translate everything that is said into a geometric statement (the trick is to realize every time talk about a field, they are really talking about the ring of integers in that field).