Best wireless router ( Apple AirPort Extreme vs Belkin N Series)

I have been advised to get a better router (read more powerful) , but am confused what to get, since my prime objective is to have seamless connectivity throughout the house.

It might be helpful to remember that the broadcast power of wireless routers is limited by law, so devices from one manufacturer to another typically perform about the same in this regard.

There is no assurance that an AirPort Extreme might perform better than your current router. The only way to know is to try it out in your home.....so it might be a good idea to understand the store's return policy in advance on any router that you might consider.

do you think that range extenders are the best choice ?

They are the only choice. It is much better to connect a range extender to the main router using an Ethernet cable than trying to use wireless only, since so much of the signal is lost using wireless.

Is there any other advantage to using an Airport over a Belkin

I am not familiar with your Belkin model, so I cannot say. Check the Apple documentation for more details on the AirPort Extreme for additional features that your Belkin product might not have.

AirPort Extreme

I do not like the clutter of wires and hence looking for a wireless solution to the issue.

Often, you must make a choice between performance (wired connections) and convenience (wireless) with networks.

I am sorry, but I doubt that another wireless extender will perform any better than the device you already have if you plan to use wireless only. Perhaps another user will post with a success story with a product that they have used.

Apple products would not be compatible for this purpose unless you already have another Apple router producing your wireless network.