Scuba diving equipment can be expensive, such as BCD's or regulators; so if you don`t dive very often renting may be the most cost-effective option. However, having your own personal equipment you are comfortable in, will improve your diving (buoyancy, air consumption, etc)
PADI is just a dive agency (certification issuer), just like RAID. Our RAID certifications are recognized worldwide, so you will never have a problem diving.
Your regulator is an expensive and obviously critical piece of equipment, if possible it is recommended to hand carry it onto flights to avoid damage. If you do not have any hand carry allowance, it can be checked in; but make sure it is well protected. Such as wrapping it in your wetsuit, and placing inside you BCD.