A mask and a cylinder are both scuba equipment, but owning them solves very different problems. One needs to fit your face reliably; the other needs to be filled, inspected, stored, and transported. Someone diving at a local quarry every weekend will make different choices from someone who spends a week in the Caribbean once a year. I wouldn't tell either diver to buy a complete kit simply because they've earned a certification.
Two Things I'd Own From the Start
I think every certified diver should own a mask and a dive computer, regardless of how often they dive. A mask needs to fit your face, and knowing it seals comfortably is worth far more than whatever you might save by renting one. Most divers already own one from their Open Water course.
The computer matters because you need to understand its display, alarms, gas settings, and controls before getting underwater. I'd rather see someone use an inexpensive computer they know well than an impressive rental whose menus they're learning on the boat.
A Wetsuit Is Personal; Fins Less So
If you regularly dive in a wetsuit, I'd strongly encourage buying one. Rental suits may be worn and still damp from the previous user, but fit is the bigger issue. A suit that circulates too much water won't keep you as warm as one that fits properly, and an overly tight suit can restrict movement.
That recommendation doesn't mean buying a collection of suits for every destination. An occasional vacation diver might own a wetsuit suited to their usual diving and rent something different when conditions demand it. Fins are less compelling to own, especially for someone who dives only on trips. If the rental fins fit and work for the conditions, there's little reason to carry your own halfway around the world.
Regulators and BCDs Depend on How Often You Dive
For regular divers, I think owning a regulator and BCD makes sense. You control their condition and maintenance and become familiar with exactly how they're arranged. You know where the BCD's dump valves and releases sit, how the regulator hoses are routed, and where the alternate second stage is located. That familiarity removes decisions you'd otherwise have to make each time you're handed unfamiliar equipment.
A diver who takes one warm-water vacation a year can reasonably rent both. The cost of ownership includes servicing, storage, and getting everything to the dive site, not merely the price on the shelf. Renting isn't an excuse to skip the checks, however. Understand the configuration, confirm the fit, and perform a proper pre-dive check before entering the water.
Cylinders and Weights Can Remain Rentals
Even frequent divers don't necessarily need to own cylinders or weights. On vacation, dive operators generally supply both, and traveling with your own is rarely practical. At home, cylinders may be worth owning when rentals or fills are inconvenient, but you also take on inspections, maintenance, storage, and transport.
Weights are a simpler decision. Owning them can make local dive preparation easier, while renting them is perfectly sensible when they're readily available. Neither purchase is a measure of experience or commitment.
Buy for the Diving, Not the Gear Locker
If you dive locally and often, owning familiar equipment usually earns its keep. If you dive once a year on vacation, there's no reason to feel inadequately equipped because you arrive with a mask and computer and rent most of the rest. You can also change that balance as your diving develops; ownership isn't a milestone you need to reach.
I own a substantial amount of equipment because I dive frequently and enjoy setting up a consistent configuration. Those are my reasons, not a prescription for someone else's diving. As I discuss in How I'd Buy Scuba Gear If I Were Starting Over, the worthwhile purchase is the one you understand and will actually use.