I have Verizon FiOS with the Actiontec MI424WR router.
The router is located in the living room, and my bedroom is on the other side of the house. The signal is much stronger in the living room than in my room where it can be virtually nonexistent at times. What I want to do is purchase either a wireless router or bridge (whichever is needed), and run an Ethernet cord from the FiOS router to my bedroom router/bridge and then create another private network separate from the house network. Is this possible without impacting or weakening the signal of the house network? Theoretically, it shouldn't...correct?
Most importantly, would I have to log into the FiOS router and change any settings? And what devices should I be looking into in order to make this feasible.
You could get the Actiontec Extender but I think you could get better bang for you buck with something like the Asus RT-N56U. You do need to wire an ethernet cord which may be a pain but you'll be upgrade ready later if you end up wanting to do any home networking, get another type of router, switch from Verizon, ect.
Option 2, a wireless bridge (such as the Netgear WNCE4004), may work; but will do nothing to improve the state of affairs for wireless devices. In my experience, the WiFi range on the Actiontec router is pretty lousy. It's also limited to 130 Mbps, which just isn't that fast by modern WiFi standards (though it is likely to be plenty fast for anything you're doing on the Internet). I believe the WCB3000N, mentioned above, improves the speed as well as the range.
That brings us to option 3. You can buy an access point and disable the Verizon router's WiFi radio. You could buy a WiFi router (as VUser50 suggests); but you'd likely just end up using it as an access point. (Unless the Verizon router's non-WiFi capabilities have proven inadequate for your needs, adding another router to your network will add unnecessary complexity.) Most any access point/WiFi router you buy at this point is likely to have better range and more recent WiFi technology than what's available in the Verizon router. But in the event that you still are looking to extend the WiFi range at that point, just about any range extender should work. The potential complication here is that the access point/router will need to connect to your Verizon router via ethernet; that might be inconvenient if you don't care to have the access point colocated with the router.
At $75, the Actiontec WCB3000N is a pretty cost-effective solution; and people who have them seem to like them. It's almost certainly the simplest solution in terms of configuration. But if you want something that supports the latest WiFi standards (802.11ac), you'll need to get an access point.