Hi M.,
While it's always good to carry chains when you are in snow country, it's not usually necessary. When we take our non-4-wheel drive vehicle up we just watch the weather carefully. For the hundreds of times I've made this journey, I've only had to use chains a handful of times. If it's going to snow (at all) then be sure to grab some chains before you go. You'll pay an arm and a leg to buy them up there.
As for resort recommendations, it's really about what kind of terrain you're looking for. The less expensive options would be Diamond Peak or Boreal, but I would definitely recommend Alpine Meadows or Northstar (although Northstar tends to be crowded), as their hills are larger and with more challenging terrain, and their VERY family friendly. And you're looking for more of a challenge, Squaw is the best - although it's also the most expensive.
Have a wonderful time!