You can do it all yourself, but if it's your first time then I'd seriously suggest you use a package tour operator, then all you have to worry about is making sure you arrive at the airport on time with your ski clothes and insurance - don't rely on just an E111 card, it may cover medical costs but will not pay the fees to get you off the mountain. They'll handle the transfer to the resort, which is often a few hours away, and help sort out your kit rental etc.
Self catering is the cheapest, but I'd suggest looking at catered chalets, it'll cost more up front but often work out cheaper as you'll not be eating out in a restaurant every night, also the food will be more in line with your dietary requirements.
Avoid going in February, it's high season due to half term hols so you'll pay a lot more and get longer lift queues and more crowded slopes in return. January can be a real bargain, but colder and the weather less predictable, March is my favourite, but it's wise to look at higher altitude places to get the best snow conditions, particularly in France.
Last, but by no means least, socks! Make sure you get some proper ski socks, you'll be spending all day clamped into a pair of ski boots, not having proper socks can ruin the whole experience.
I've just booked (last week) a week in La Tania (3 valleys) on 11th Jan in a catered chalet from Gatwick. Grand sum of £379!!!!