The characterization and construction of bent functions are challenging problems. The paper generalizes the constructions of Boolean bent functions by Mesnager \cite{M2014}, Xu et al. \cite{XCX2015} and $p$-ary bent functions by Xu et al. \cite{XC2015} to the construction of $p$-ary weakly regular bent functions and presents new infinite families of $p$-ary weakly regular bent functions from some known weakly regular bent functions (square functions, Kasami functions, and the Maiorana-McFarland class of bent functions). Further, new infinite families of $p$-ary bent idempotents are obtained.