Ditto on "both". I wear fleece gloves - they're not quite windproof (despite the labelling), and not extremely warm, but most of the time I just want something to keep the chill off. The fleece breathes well so my hands don't get sweaty, which is as much a concern as the cold. Dexterity is much better than with mittens - I can usually manage zippers, crampons, etc without removing the gloves.
Above treeline on the coldest days, I'll put a pair of nylon mitts over the gloves for a completely windproof, waterproof shell that extends over my jacket sleeve to the elbow.
Consider bringing more than one pair of fleece gloves - they're not waterproof, so if you fall into the snow, or spill your water, or just sweat a lot, the gloves you're wearing will get wet. Not only does that make them less warm, but if you take them off when wet they can freeze solid.