I have tracked down some results on explicit classifications of simple modules for $u_q(\mathfrak{sl}_3)$. The general picture is that the simple modules are bigraded by the root lattice and look like towers of concentric hexagons.
For the benefit of anyone else interested, there is a long series of papers by Dobrev:
Dobrev, V. K., Multiplet classification of highest weight modules over quantum universal enveloping algebras: The $U_ q(sl(3,\mathbb{C}))$ example, Groups, Vol. 1, Proc. Int. Conf., St. Andrews/UK 1989, Lond. Math. Soc. Lect. Note Ser. 159, 87-104 (1991). Zbl 0758.17008.
Dobrev, V. K., Representations of Quantum Groups, Gruber, B. (ed.) et al., Symmetries in Science V. Springer, Boston, MA (1991).
Dobrev, V. K., Classification and characters of $U_q(sl(3,\mathbb{C}))$ representations, Quantum groups, Proc. 8th Int. Workshop Math. Phys., Clausthal/Germ. 1989, Lect. Notes Phys. 370, 107-117 (1990). Zbl 0727.17004.
Dobrev, V. K.; Truini, P., Irregular $U_q(\mathrm{sl}(3))$ representations at roots of unity via Gel'fand–(Weyl)–Zetlin basis, J. Math. Phys. 38, No. 5, 2631-2651 (1997). Zbl 0965.17009.
Dobrev, V. K.; Truini, P., Polynomial realization of the $\mathrm{U}_ q(\mathrm{sl}(3))$ Gel'fand–(Weyl)–Zetlin basis, J. Math. Phys. 38, No. 7, 3750-3767 (1997). Zbl 0882.17005.
Dobrev, V. K., Characters of the $U_ q(sl(3,\mathbf{C}))$ highest weight modules, Eguchi, T. (ed.) et al., Common Trends in Mathematics and Quantum Field Theories. 1990 Yukawa international seminar school: Kansai Seminar House, Kyoto, Japan, May 10-16, 1990. Workshop: RIMS, Kyoto University, Japan, May 17-19, 1990. Tokyo: Yukawa Institute for Theoretical Physics, Prog. Theor. Phys., Suppl. 102, 137-158 (1990). Zbl 0784.17018. MR1182163.
Dobrev, V. K., Representations of quantum groups for roots of $1$, Domokos, G. (ed.) et al., Nonperturbative Methods in Low Dimensional Quantum Field Theories (Debrecen, 1990), 69–105, World Sci. Publ., River Edge, NJ, 1991. MR1191199.
…and many others.
Also there is a paper by Abdesselam, Arnaudon, Chakrabarti:
- Abdesselam, B.; Arnaudon, D.; Chakrabarti, A., Representations of ${\mathcal U}_q(sl(N))$ at roots of unity, J. Phys. A, Math. Gen. 28, No. 19, 5495-5507 (1995). Zbl 0864.17016.
and a discussion of dimensions by Mariana Pereira here:
Some of the relevant material is hard to find and/or requires paying large sums of money to publishing corporations.
