Veronese rings, Segre embeddings, or more generally Segre--Veronese embeddings are very important rings in algebraic geometry. In this paper we present an original, elementary way to compute the Hilbert--Poincar\'e series of these rings; as a consequence we compute their Castelnuovo--Mumford regularity and also the highest graded Betti number. Moreover, using the Castelnuovo--Mumford regularity of a Cohen--Macaulay finitely generated graded module, we compute that of its Veronese transforms.