The pros and cons for owning your own router varies for every situation, but in general, the following usually counts:
- Cost savings (Eliminating the monthly router rental fee from Verizon - often times in the long run using your own router can save you quite a bit)
- For techies, more advanced control over your local network
- Depending on your situation, better performance than Verizon's router
- Older FiOS installations require you to run an ethernet cable to your optical network terminal, which is usually located in an inconvient part of the house.
- Some FiOS specific features may be lost such as Caller ID on TV, remote DVR, etc.
Which router to purchase relies heavily on your specific needs. If you're looking for a simple router to just replace the Verizon router, most generic off-the-shelf routers will suffice. Personally, I'm a fan of TP-Link's products for a simple, no-nonsense router.
Another option would be planning out your network beyond a simple router. Often times, the places where our router lives isn't actually the best location for optimal WiFi signal. If this is the case, you could purchase a hardwired router (I prefer the EdgeRouter X from Ubiquiti as the most feature-rich for the price), and have separate WiFi Access points (such as the Unifi AC Lite) spread out throughout your house for best performance. This is usually a good option if you're in the process of a new home build (or renovation) where running cabling is an option.
Lastly, if you're just looking to get better WiFi coverage than the normal Verizon router, you could look into Unifi's plug and play Amplifi system. This system creates a wireless mesh network throughout your house for maximum WiFi strength. Although personally I would definitely suggest to hardwire your WiFi access points if at all possible. WiFi is a shared spectrum, so the more devices you have repeating the signal, the more noise is generated, and the worse your experience will be. Mesh systems are great if you live in the suburbs with some distance between you and your neighbor's router, but if you live in a dense urban environment, I'd advise you to avoid mesh systems.
Please do let me know if you have any more questions