In reply to timcook207: Pretty decent answers already.
For grade I winter routes of whatever variety you should never need anything more than what you already own. For the vast majority, a rope and a couple of slings will suffice. Nuts generally won't be need, but you lose little carrying some. An old sling or some tape or 5/6mm cord as abseil tat is about the only other thing you might want.
For mountaineering style, grade II routes (such as the Aonach Eagach or Forcan Ridge) pretty much exactly the same applies. One axe will suffice but some mountaineers still want to carry a second 'for peace of mind' or added flexibility. Just remember that deciding to carry one is no justification for deploying it immediately when you rope-up. On moderately-angled snow/ice covered rock, trying to use a second axe is far more often a liability than a help. (In three winter ascents of Tower Ridge, I have yet to use a second axe but I must have witnessed a dozen climbers making a hash of trying to climb sections of it with two.)
However, when it comes to the shorter climbs (mainly gullies) at popular venues such as Aonach Mor, The Northern Corries or Stob Coire nan Lochan things are rather different. A second axe is pretty much essential on every route from grade II upwards. You will also start to need a larger rack. A set of nuts and a couple of hexes will serve for most grade II routes and it is only once you reach grade III that any of warthogs/bulldogs, pitons or ice screws become appropriate depending on the route and conditions.