A finite products of metrizable spaces is metrizable. A discrete topology on R is metrizable since it is induced by a discrete metric on R. A standard topology on R is also metrizable since it is induced by a standard metric on R. You can also use some metrization theorems here. You can use a Nagata-Smirnov metrization theorem to show that a discrete topology on R is metrizable and use a Urysohn metrization theorem to show that a standard topology on R is metrizable.