Walnut - where do you live at the moment? If you've never been a Londoner, the thing to understand is that nice places and not so nice places are very squished together - sometimes by as little as a couple of roads. West Greenwich isn't particularly near New Cross though - I would say it's near Deptford and Lewisham.
MV is ok, but as others have said, a bit lacking in soul. There are no real amenities there and you're a reasonably substantial walk from Greenwich centre (although there is a bus).
Fundamentally, it will come down to budget. You will get a lot more for your money in East Greenwich rather than West. I would also actually prefer to commute from East Greenwich to CW. As others have said, it's a bus down to the Jubilee line. However, the DLR can be mighty, mighty slow. Once you are on the tube it's only one stop, so although you might spend a while getting to the station, you hardly spend any on the tube.
East Greenwich is within walking distance of the tube if you're in the smaller houses off Trafalgar Road (Whitworth, Pelton, etc - have a look on Google maps) or down Blackwall Lane (Azof, etc). About 20 minutes at a fast walking pace. Bit harder if you're further up the hill in the larger houses. West Greenwich isn't walking distance to the tube IMO. It's about 15/20 minutes more, depending on where you are coming from. However, West Greenwich is much closer to the DLR. If you are looking about transport in and out of town (not just to CW), also bear in mind that East and West Greenwich both have train stations (Greenwich, then Maze Hill and Westcombe Park for the east.)