This paper is devoted to establishing several formulas relating Bernoulli numbers and Stirling numbers of both kinds. Some of these formulas are rediscoveries, presented with new proofs or from a fresh perspective, while others are entirely novel. Among the key results, we express Bernoulli numbers of the first kind in terms of Stirling numbers of the second kind, and another expresses Bernoulli numbers of the second kind in terms of Stirling numbers of the first kind. Additional formulas provide summation identities mixing Stirling numbers of both kinds with Bernoulli numbers (either of the first or second kind). Finally, the most original results transform certain linear combinations of Bernoulli numbers (first or second kind) into linear combinations of Stirling numbers (first or second kind).