If I own a patent, do I have the right to exploit the invention?

A patent gives you the right to exclude others from practicing your invention. But if practicing your invention requires that you practice another's patented invention, you could be held liable for infringing the other patent.

No, your patent does not give you the right automatically to practice your own invention - and you certainly do not have the right universally - patents are territorial - each country or group of countries controls the patent rights. The patent gives you the exclusive right to prevent others from making, using, or selling your patented invention but does not give you that right because if you choose to use your patented technology, you may possibly infringe other patents.

First, I've seen many people think they got a patent when all they'd done was file or provisional or while their utility was under examination.

Second, "universal" is a broad word. No, you can only practice the patent with protection (subject to constraints stated below) in the jurisdictions in which you've been issued a patent. Often, that's just the US. If you file under the PCT (Patent Cooperation Treaty), you can apply for patents simultaneously in over 100 countries, but in general, each country does a separate examination. It's complicated. Bottom line is a US Patent doesn't give you anything in Canada, let alone Europe, Asia or anywhere else.

Third, "universal" seems to suggest you can do anything. You can't. Your patent, if issued, is different from other patents. They are protected IP, too. So if you wish to include any art described in other patents, then you need to negotiate a license to be able to use that IP. Otherwise, it's you who is infringing.

Before to exploit your patented technology you need to be sure that you will not infringe third party patents in that territory, therefore it is higly recommended to carry out a freedom to operate analysis before such exploitation.