Will any wireless router work with tablets?

You need to make sure it's an ethernet input router, generally called a "cable router" or "DSL Router".

And NOT a telephone line input "ADSL Router" or "modem router" - these are for ADSL brodband only, and cannot connect to a cable modem.

The Kindle Fire supports 802.11b/g/n, but does not have particularly fast Wifi (it's upgraded on the HD, to dual band and MiMo).

For the Fire, a router capable of N-150 would be adequate, and unless other systems are more capable, it would be best to set it to 20MHz mode (no extension channel), Routers with N-300 capability would be compatible, but may not offer any greater performance.

If it's a Fire HD, the the more basic routers would still work, but it would potentially have better performance with an N-300 or dual band - the 5GHz band is much less cluttered, so has better perfomance versus interference/high occupancy, but suffers more from obstruction, while the more common 2.4GHz band is cluttered, so using 40MHz (extension channel) is somewhat antisocial.

Pretty much. Looks like it supports 802.11 abgn, which means I think you would have to look REALLY hard for a wireless router that wasn't basically compatible with!

Of course, MOST of the time, these things just work pretty much right out of the box, but unless someone specifically gripes about the router you're going to buy online, there's really know way to know 100%. It's always a little bit of a crap shoot, some routers just suck, and some devices do too.

WiFi is WiFi. Any normal N300 router will do. Depending on where you're dwelling, you would to search out $30 wi-fi router, however efficiency and sturdiness may go away anything to be appreciated,