The Catalan sequence is a very important sequence of integers with many combinatorial interpretations. In this work we are interested in exploring patterns of prime divisibility within this sequence using the notions of p-adic valuations and primitive prime divisors. We explicitly describe the pattern of the 3-adic valuation of the Catalan numbers in the context of Alter and Kubota's description of the p-adic valuation of the Catalan numbers for p>3. Then we use these results and Alter and Kubota's work to determine the growth of the p-adic valuation which informs us about the growth of multiplicities of prime divisors of the Catalan numbers. Moreover, we extend this result to p=2 despite the fact that the 2-adic valuation of the Catalan Numbers is almost always treated separately. We also study the primitive prime divisors of the Catalan sequence and determine its Zsigmondy set. We additionally study other sequences related to the Catalan numbers and explore how the arithmetic properties of these sequences are similar or different from that of the Catalan numbers. In particular, we look at the 2-adic valuation of the partial sums of the Catalan numbers and see that these numbers have a much different arithmetic structure. We remark on the k-Catalan numbers and the ballot numbers which vaguely share some arithmetic patterns. Finally, we study the primitive prime divisors of the super Catalan numbers, the central binomial coefficients, and the convolutions of the Catalan numbers and see that these sequences are arithmetically very similar to the Catalan numbers.