To add another viewpoint to what Paul has already said, it's important to realize that a lot of general theory exists by now for these particular groups and for others of Lie type (especially the work inspired by Lusztig). The Atlas authors relied on only some of this work, but started with older literature on special cases and some direct computer work for large groups like the Monster.
But for finite groups of Lie type there is an extensive theory to tap into, beyond the few special cases treated in earlier literature. Paul has indicated some of the ideas coming from the 1955 paper by J.A. Green in Trans. Amer. Math. Soc. Green treated the somewhat easier case of finite general linear groups, using combinatorial methods. But special linear groups are harder to work with, due to the finite center in the ambient algebraic group. Here the explicit character values have only recently been made fairly explicit for cases beyond $SL_3(\mathbb{F}_q)$.
The limited case you start with involves (as do other groups of Lie type) some exceptions for small primes, but when $p$ is large enough the picture becomes rather uniform. Two specific references deal with the questions you raise:
1) Before the "modern" theoretical era, W.A. Simpson and J.S. Frame worked out character tables for some rank 2 groups including $SL_3(\mathbb{F}_q)$ for a power $q$ of $p$. Their results are mostly reliable, with a little bit of adjustment, and published in Canad. J. Math. 25 (1973): PDF here.
In particular, the largest degrees occur for principal series characters (as Paul indicates). Here as in other families of Lie-type groups there are standard degree polynomials in $q$ whose highest term is always $q$ raised to the number of positive roots (in your example 3). But for small primes all of these families of characters may be degenerate, preventing the generic degrees from occurring in the character table. The details do get complicated, but keep in mind that as $p$ grows the character tables become more uniform and predictable. There are lots of references, but the bottom line is that much is known. Not everything, of course.
2) Quite a few people have worked out the total number of classes (hence of complex irreducible characters) for families of Lie type groups. I think the joint paper by Deriziotis and Holt in Comm. Algebra 21 (1993) might contain the most explicit results for the next-larger family $SL_4$. In your case $SL_3$ it may be easiest to compile the number of characters from the Simpson-Frame paper linked above. Roughly speaking, the total number is given by a polynomial in $q$ of degree equal to the Lie rank (in your example 2).
ADDED: If my arithmetic is correct, the number of classes (or characters) for $SL_3(\mathbb{F}_q)$ with $q$ a power of $p$ is $q^2+q$ when $d=1$ and $q^2 + q + 8$ when $d=3$. Here $d = \gcd(3,q-1)$, the order of the center.
One other remark: the Atlas gives little help with questions about asymptotic behavior as $p \rightarrow \infty$, but a comparison with $p$-modular representations may help to "explain" the significance of the asymptotic values given by highest powers in polynomials above. Here $p^2$ is the number of modular irreducibles, while $p^3$ is the dimension of the largest of these (the Steinberg representation). See for instance J. Algebra 72 (1981), 8--16.
