I am getting grips and gloves. My instructor said get both and they are out all day everyday.
The Keis 601 are battery and 12v which I thought covered all bases.
I often rode all day in temperatures down to -10ºC, which at (legal) motorway speeds gives a wind chill around -25ºC... Not fun on an unfaired Cruiser, lemme tell you!
Keis are a decent brand, from what I've read in general, but can be rather expensive.
Gerbing used to be one of the top brands. Ixon are another to consider, too.
The first problem with heated grips is that, being the bit you hold for control of the bike, when you decide they've gotten too hot you're then stuck holding what can often be VERY hot grips until the wind can cool them down. Since your hand insulates them, that can take a dangerously long time. By contrast, once you turn them down/off, the wind tends to cool the gloves much faster.
The second problem with what your instructor has said is the high current draw of two things can be a problem for some bikes.
The third problem is that heated grips are usually thicker than standard grips. Some people really don't get on with this, especially when wearing thick winter gloves. Remember this is your primary point of control on the bike, and in wintery conditions that can matter a lot.