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In this article, we theoretically study, in an atomic-optomechanical system, quantum correlations shared between three modes, namely mechanical mode, optical mode and atomic mode. We firstly investigate the combined effect of the variation of the cavity-collective atomic mode coupling and the effective optomechanical one, on the tripartite entanglement behavior as well as on tripartite steering evolution. Then, we discuss pairwise entanglement and bipartite steering according to the aforementioned couplings. Consequently, besides monogamy of the entanglement distribution, the sharing of Gaussian steering is also monogamous, thus the CKW-type monogamy inequalities are fully satisfied under all permutations of the three considered modes in a chosen stable region. In addition, the study of tripartite entanglement and tripartite steering behaviors leads to know the optimal conditions to generate genuine tripartite entanglement, one-way and two-way steering. [ABSTRACT FROM AUTHOR]