In this paper, we establish several properties of the unified generalized Stirling numbers of the first kind, and the Jacobi-Stirling numbers of the first kind, by means of the convolution principle of sequences. Obtained results include generalized Vandermonde convolution for the unified generalized Stirling numbers of the first kind, triangular recurrence relation for general Stirling-type numbers of the first kind, and linear recurrence formula for the Jacobi-Stirling numbers of the first kind, and so forth, thereby extending and supplementing known knowledge to the existent literature about these Stirling-type numbers.