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](http://theo.inrne.bas.bg/~dobrev/):

- <cite authors="Dobrev, V. K.">_Dobrev, V. K._, [**Multiplet classification of highest weight modules over quantum universal enveloping algebras: The $U_ q(sl(3,\mathbb{C}))$ example**](https://doi.org/10.1017/CBO9780511721236.012), Groups, Vol. 1, Proc. Int. Conf., St. Andrews/UK 1989, Lond. Math. Soc. Lect. Note Ser. 159, 87-104 (1991). [Zbl&nbsp;0758.17008](https://zbmath.org/0758.17008).</cite>

- <cite authors="Dobrev, V. K.">_Dobrev, V. K._, [**Representations of Quantum Groups**](https://doi.org/10.1007/978-1-4615-3696-3_5), Gruber, B. (ed.) et al., Symmetries in Science V. Springer, Boston, MA (1991).</cite>

- <cite authors="Dobrev, V. K.">_Dobrev, V. K._, [**Classification and characters of $U_q(sl(3,\mathbb{C}))$ representations**](https://doi.org/10.1007/3-540-53503-9_44), Quantum groups, Proc. 8th Int. Workshop Math. Phys., Clausthal/Germ. 1989, Lect. Notes Phys. 370, 107-117 (1990). [Zbl&nbsp;0727.17004](https://zbmath.org/0727.17004).</cite>

- <cite authors="Dobrev, V. K.; Truini, P.">_Dobrev, V. K.; Truini, P._, [**Irregular $U_q(\mathrm{sl}(3))$ representations at roots of unity via Gel'fand–(Weyl)–Zetlin basis**](https://doi.org/10.1063/1.532001), J. Math. Phys. 38, No. 5, 2631-2651 (1997). [Zbl&nbsp;0965.17009](https://zbmath.org/0965.17009).</cite>

- <cite authors="Dobrev, V. K.; Truini, P.">_Dobrev, V. K.; Truini, P._, [**Polynomial realization of the $\mathrm{U}_ q(\mathrm{sl}(3))$ Gel'fand–(Weyl)–Zetlin basis**](https://doi.org/10.1063/1.532074), J. Math. Phys. 38, No. 7, 3750-3767 (1997). [Zbl&nbsp;0882.17005](https://zbmath.org/0882.17005).</cite>

- <cite authors="Dobrev, V. K.">_Dobrev, V. K._, [**Characters of the $U_ q(sl(3,\mathbf{C}))$ highest weight modules**](https://doi.org/10.1143/PTP.102.137), 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&nbsp;0784.17018](https://zbmath.org/0784.17018). [MR1182163](https://mathscinet.ams.org/mathscinet-getitem?mr=1182163).</cite>

- <cite authors="Dobrev, V. K.">_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](https://mathscinet.ams.org/mathscinet-getitem?mr=1191199).</cite>

…and many others.

Also there is a paper by Abdesselam, Arnaudon, Chakrabarti:
- <cite authors="Abdesselam, B.; Arnaudon, D.; Chakrabarti, A.">_Abdesselam, B.; Arnaudon, D.; Chakrabarti, A._, [**Representations of ${\mathcal U}_q(sl(N))$ at roots of unity**](https://doi.org/10.1088/0305-4470/28/19/007), J. Phys. A, Math. Gen. 28, No. 19, 5495-5507 (1995). [Zbl&nbsp;0864.17016](https://zbmath.org/0864.17016).</cite>

and a discussion of dimensions by Mariana Pereira here:
- [Dimensions of simple $u_q(\mathfrak{sl}_3)$-modules](http://www.cmat.edu.uy/~mariana/dobrev.pdf).

Some of the relevant material is hard to find and/or requires paying large sums of money to publishing corporations.