Leave the router connected at all times. The latest routers are constantly negotiating the best data exchange rate and responding to noise and interference on the connection.
Changing weather conditions can alter the residual background noise levels on the line, particularly with overhead cables.
Over a period of time, your router "learns" the best coping category to deal with noise on the line and maintain the highest connection speed to the exchange. If you turn off the router, this information is reset and the router has to relearn the best settings all over again. This can indeed take several days to establish an otimum connection again, particularly if you are served by overhead lines and are distant from the nearest exchange.
Power consumption is only a few watts, no more than leaving an automatic LED nightlight turned on constantly. Any money saved by turning off the router will be swallowed up by the frustration of longer upload and download times for data and the need to have your computer turned on for longer to access that data.
Plug it in, set it up and forget it.
Until something goes wrong...