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).
Let's start with the most elementary example: projective space $\mathbb P^n$. It's not hard to see that that the number of points on it is always $q^n + q^{n-1} + \dots + q + 1.$
Note that this is because $\mathbb P^n$ can be always decomposed into simpler pieces: $\mathbb A^n \cup \mathbb A^{n-1}\cup\dots\cup \mathbb A^0$. Interestingly, something similar applies to all $\mathbb F_q$-varieties. Specifically, the Lefschetz fixed points formula from topology applied to arithmetics gives the following statement for a variety $X/\mathbb F_q:$
There exist some algebraic numbers $\alpha_i$ with $|\alpha_i| = q^{n_i/2}$ for some $(n_i)$ such that the number of points $$\\# X(\mathbb F_{q^l}) = \sum_i (-1)^{n_i}\alpha^l_i\quad \text{for}\\ l > 0 .$$
Numbers $\alpha_i$ in fact come from geometry: they are eigenvalues of some operators acting on etale cohomology groups $H_{et}(X)$. In particular, the numbers $n_i$ can only occupy an interval between 0 and $\text{dim}\\, X$ and there are as many of them as the dimension of this group.
These groups can directly compared to the case of $\mathbb C$ whenever you construct your variety in a geometric way. To see how, consider the example of curves. Over $\mathbb C$ the cohomology have the form $\mathbb C \oplus \mathbb C^{2g} \oplus \mathbb C\ $ for some $g$ called genus; the same holds over $\mathbb F_q$:
- projective line $\mathbb P^1$ has genus 0, so it always has $n+1$ points
- elliptic curves $x^2 = y^3 + ay +b$ have genus 1, so they must have exactly $n + 1 + \alpha + \bar\alpha$ points for some $\alpha\in \mathbb C$ with $|\alpha| = \sqrt q.$ This is exactly the Hasse bound mentioned in another post.
These theorems, which provided an unexpected connecion between topology and arithmetics some half-century ago, were just the beginning of studying varieties over $\mathbb F_q$ using the geometric intuition that comes from the complex case.
You can read more at any decent introduction to arithmetic geometry or étale cohomology. There are also some questions here about motives which are a somewhat more abstract version of the above picture.
As a reply to Ben's comment above about reconstructing the genus if you know $X_n = \#X(F_{q^n})$:
You know with certainty that $1 + q^n - X_n = \sum \alpha_i^n\ $ for some algebraic numbers $\alpha_i, i = 1, 2, \dots $ having property $|\alpha_i| = \sqrt q.$
There cannot be two different solutions $(\alpha_i)$ and $(\beta_i)$ for a given sequence of $X_n$ because if $N$ is a number such that both $\alpha_i = \beta_i = 0$ for $i>N$ then both $\alpha$ and $\beta$ are uniquely determined from the first $N+1$ terms of the sequence.
So a given sequence uniquely determines the genus.
I don't know, however, if a constructive algorithm that guarantees to terminate and return genus for a sequence $X_n$ is possible. The first idea is to loop over natural numbers testing the conjecture that genus is less then $N$, but there seem to be some nuances.
Best Answer
The Diophantine equation $x^2 - 34y^2 = -1$ has no integer solutions, even though it has solutions in ${\bf Z}_p$ for all $p$ (including $p = \infty$ if we understand "${\bf Z}_\infty$" as $\bf R$). This is the first example of the failure of the Hasse principle for the minus case of the Fermat-Pell equation $x^2 - \Delta y^2 = \pm 1$ (with $\Delta $ a fixed positive integer that is not a square), or equivalently for the existence of units of norm $-1$ in ${\bf Z}[\sqrt{\Delta}]$. It can also be regarded as the first example of a nontrivial element of the "Tate-Šafarevič group" for the torus $x^2 - \Delta y^2 = +1$ (since $x^2 - \Delta y^2 = -1$ is a principal homogeneous space for that torus).
[NB the equation $x^2 - 34y^2 = -1$ does have rational solutions, such as $(x,y) = (5/3,1/3)$. Indeed Minkowski already showed that a quadratic equation in any number of variables has a rational solution iff it has a solution in each ${\bf Q}_p$ and in ${\bf R}$; Hasse generalized this from ${\bf Q}$ to an arbitrary number field.]
[Added later:] In general $x^2 - \Delta y^2 = -1$ has solutions in every ${\bf Z}_p$ iff $\Delta$ is either a product of primes congruent to $1 \bmod 4$ or twice such a product; equivalently, iff $\Delta$ is the sum of two coprime squares. If such $\Delta$ is of the form $n^2 \pm 2$ then $(n + \sqrt\Delta)^2 / 2$ is a unit of norm $+1$, and is fundamental unless $\Delta=2$. This accounts for infinitely many examples, including the first two, $\Delta = 34 = 5^2 + 3^2 = 6^2 - 2$ and $\Delta = 146 = 11^2 + 5^2 = 12^2 + 2$ (see OEIS sequence A031398). The infinitude may be shown with a polynomial identity such as $$ (2t^2+2t+1)^2 + (2t+1)^2 = (2t^2+2t+2)^2 - 2 $$ which recovers $\Delta = 34$ for $t=1$. It's then a natural question to ask: as $M \rightarrow \infty$, among those positive $\Delta < M$ that are sums of two coprime squares, for what fraction does $x^2 - \Delta y^2 = -1$ have solutions? I guess that it is conjectured, but not known, that there is a positive limit strictly smaller than $1$.