Hello seascallop,
I'm afraid there isn't a simple answer to your question. Mostly, it's a matter of personal preference and comfort. Personally, I prefer to have as much of my own gear as possible, and certainly mask, fins, wetsuit. They don't weigh much or take up much room in the bottom of a bag, so I just bring them.
In my opinion, before I use a rental regulator, I would want to know some things about the place that I'm diving, and who I'm renting from. I used rented regs several times without a problem, but I always made a point of knowing who was doing the service on the regs, and knowing how frequently they were serviced. It's really all a question of trust.
Regulator cost can range from $350-$1000 or higher. While many people don't own regs (and are very happy that way), those who do own them will tend to tell you how great it is. I am personally a fan of the 'rent before you buy' philosophy. Find out what you like, and what you need, before you let a salesman influence your choice. And of course, while you're renting, you'll meet other divers who own their own gear, and you can ask them questions about it. Most of them are happy to talk about their setups, and don't have any financial stake in telling you what to do.
Get in the water, have fun, and be safe.