It's certainly not too hard to understand everything there is to understand about the algebraic closure of Fp. Perhaps the reason this is unsatisfying as an example for founding intuition is because it doesn't really have a nice topological structure; it lacks anything like a natural metric. So here's an attempt to explain why what is in some sense the next simplest example puts you in a better situation, intuition-wise.
If you have some intuition about the p-adic numbers look and feel (for example, topologically), then you secretly have intuition for the t-adic topology on the complete local field K=Fp((1/t)). Now, as far as characteristic p fields go, this sort of puts you in the position of (in your parlance) a "preschooler" who knows about R but hasn't yet gotten to kindergarten to learn about C. Why is K like R? First, it is locally compact. Second, it is at least analogous to completing Fp(t), which is very much like Q with Fp[t] as the analogue of Z, at an "infinite" valuation, namely the degree or (1/t)-adic valuation, rather than a "finite" place like a prime polynomial in Fp[t]. (The (1/t)-adic valuation corresponds to the point at infinity on the projective line over F_p. Likewise, number theorists love to say, perhaps partly to annoy John Conway, that the real and complex absolute values on Q correspond to "archimedean primes" or "primes dividing infinity". This is actually a pretty lame analogy, though, since K=Fp((1/t)) looks a lot more like Fp((t)), say, than R or C looks like Q2.)
Unfortunately there are two extra difficulties in the characteristic p case. First, upon passing to the algebraic closure L of K we lose completeness. Second, we make an infinite field extension, unlike the degree 2 extension C/R. Thus, while L is an algebraically closed field of characteristic p, it bears little resemblance to R. In fact, it's a lot more like an algebraically closed field of characteristic 0 that is a bit scarier (at least to me) than C, namely Cp, or what you get when you complete the algebraic closure of Qp with respect to the topology coming from the unique extension of the p-adic valuation.
While this may seem bad, I think it's actually good, because one can really get a handle on some of the properties of Cp. [Note that as another answerer pointed out, Cp = C as a field, but not as a topological or valued field, which is really a more interesting structure to consider from the viewpoint of intuition anyway.]
For example of some similarities, miraculously Cp turns out to still be algebraically closed, and I believe the same proof goes through for L above. Another property L and Cp share is that in addition to "geometric" field extensions K'/K obtained by considering function fields of plane curves over Fp, there are also "stupider" extensions coming from extending the coefficient field. This is like passing to unramified extensions of p-adic fields, where one ramps up the residue field. (In fact, it's exactly the same thing.) Both L and Cp are complete valued fields with residue field the algebraic closure of Fp. (But the valuation is NOT discrete; it takes values in Q.) There are some dangerous bends to watch out for topologically, however. Some cursory googling tells me that Cp is not locally compact, although it is topologically separable.
In addition, positive characteristic inevitably brings along the problem of inseparable field extensions sitting side L. This is, of course, an aspect where L/K is unlike Cp/Qp. Notwithstanding such annoyances, I would argue that the picture sketched above actually does give an example of an algebraically closed field of characteristic p for which it is possible to have some real intuition.
I am also just learning this stuff, and I'm partly writing this out for my own benefit. Experts, please correct and up/down vote as appropriate!
The goal of the minimal model program is to give a standard, nonsingular, representative for each birational class of algebraic variety. As stated, this goal is too ambitious, but it will help us to understand the minimal model program if we think of it as a partially successful attempt at this goal.
Let $X$ be a compact, smooth algebraic variety of dimension $n$. Let $\omega$ be the top wedge power of the holomorphic cotangent bundle. Then the vector space, $V:=H^0(X, \omega)$, of holomorphic $n$-forms on $X$ is a birational invariant of $X$. This means that we should be able to see $V$ from just the field of meromorphic functions on $X$; here is a sketch of how to do that. So we get a rational map $X \to \mathbb{P}(V^{\*})$ by the standard recipe. More generally, we can replace $\mathbb{P}(V)$ with Proj of the ring $\bigoplus H^0(X, \omega^{\otimes n})$. This is called the canonical ring; you may have heard of the recent breakthrough in proving that the canonical ring is finitely generated. We can map $X$ rationally to this Proj; the image is called the log model. This is a partial success: it is a canonical, birational construction, but it may not be birational to $X$ and may not be smooth.
There are certain well understood rules of thumb for how various subobjects of $X$ behave in the log model. For example, if $X$ is a surface and $C$ a curve with negative self intersection, then $C$ will be blown down in the log model.
Here is a more complicated example, which is relevant to your question. Let $Y$ be some variety that locally looks like the cone on the Segre embedding of $\mathbb{P}^1 \times \mathbb{P}^1$. So $Y$ is a $3$-fold with an isolated singularity. If you are familiar with the toric1 picture, it looks like the tip of a square pyramid. Inside $Y$, let $Z$ be the cone on one of the $\mathbb{P}^1$'s. This is a surface, but not a Cartier divisor. Let $X$ be $Y$ blown up along $Z$; so that the isolated singularity becomes a line. In the toric picture, the point of the pyramid has lengthened into a line segment, and two of the faces which used to touch at the point now border along an entire edge. In the log model, the line will blow back down to become a point. So the log model can turn a smooth variety, like $X$, into a singular one like $Y$.
Now, birational geometers did not rest on their laurels when they had constructed the log model. They made other constructions, which are smoother but less canonical. Many of these constructions can be thought of as taking the log model and modifying it in some way. If the log model looks like the example of the previous paragraph, they want to take the singular point of $Y$ and replace it by a line, to look like $X$. But they have two ways they can do this; they can blow up one $\mathbb{P}^1$ or the other; giving either $X$ or $X'$. Often, replacing $X$ by $X'$ is crucial in order to improve the model somewhere else. The relationship between $X$ and $X'$ is called a flip, because we take the line inside $X$ and flip it around to point in a different direction.
1 Cautionary note: although the toric picture is excellent for visualizing what is going on locally, you shouldn't take $X$ itself to be a toric variety. There are no global sections of $\omega$ on a toric variety, so the log model is empty. You want $X$ to locally look like a toric variety, but have global geometry which is nontoric in a way that creates lots of sections of $\omega$.
Best Answer
In the surface case, MMP in char p is known. See Koll'ar-Kov'ac's preprint on Koll'ar's webpage.
In dimensional 3, the existence of divisorial contractions and flipping contractions is known as EWM (so the target is only known as a algebraic space). See Keel's paper BASEPOINT FREENESS FOR NEF AND BIG LINE BUNDLES. I'm not sure about the termination of flips. The existence of flips is certainly not known (at this moment).
In higher dimensions, I think almost nothing is known.