Welcome to broadband!
It's described as "always on" .... and that's how it prefers to be. The leccy consumed by a router is pretty insignificant ... be tempted to leave it on ... you can switch the rest of your kit off.
If you are determined - start with BT 151 (assuming they provide your line.
get that tested ....
if OK ... it's almost certainly the MTU in the exchange that's lazy - but getting the equipment checked will probably require a few hours concerted effort as you get the good old dsl run-around
line fault ... must be the modem .... unplug your phone .... re-install... take the router back etc etc ... you know how it goes
if you don't you will by the time you convinge someone to do anything ....
good luck