Big Ben Club
La Tzoumaz is really easy to get to - IF you have a car. The road up is steep with many turns, however it tends to be open the entire season, rarely is it shut due to snowfall. It is much easier to get here by car, although the other options are below. It is also JUST off the motorway after Martigny so after exiting the motorway only about 15-20 minutes up the mountain.
There are many rental options available at Geneva airport, you can book online. Beware that the Swiss make you buy a "vignette" basically a sticker for your windscreen that costs 40 CHF if you drive on the Swiss motorways. The fines are high if you get caught without one. Geneva to La Tzoumaz is about 2 hours. There are many speed cameras between Geneva and Lausanne. If you do more than 160kph on the Swiss motorways, then you could potentially go to jail....
Drive to Riddes (just after Martigny) and then follow the signposts to La Tzoumaz (it is signposted, follow the signs to Riddes from the motorway, then La Tzoumaz is signposted to the right (before a furniture shop). If you have a front wheel drive car, it would be good to have winter tyres and snow chains (in case). If you have a rear wheel drive car (BMW), then be careful as the snow chains will not give you much control.
Train to Riddes
The Swiss train service is generally excellent, but it can be expensive, but you have a few options to make it cheaper.
Regular fare: Geneva Airport to Riddes - one way 50CHF
With 1/2 price card: 25CHF, but the 1/2 price card is ~175CHF per year
With 1/2 price card: 25CHF, but the 1/2 price card is ~60CHF per month
Super saver tickets - restricted to a certain train at a certain time, reserve in advance
From Riddes to La Tzoumaz
With public transport, there are a couple of options. If you are lucky, then you can get a bus directly up to La Tzoumaz. If unlucky, then you need to catch the telecabine from Riddes to Iserables and then the bus (from outside the door) to La Tzoumaz.
You can also order a taxi to share - Jackie is one of the local guys that will get you to the resort for about 40-50 CHF (I'll add the contact when I get his number).
If you are staying at a local hotel, you might be able to get them to send a driver down for free if you ask nicely.
You can get the train from Geneva to Chable (just below Verbier) and then pay to get the telecabine up from the roundabout to Savolyeres and then down again to La Tzoumaz (last lift up from Verbier at ~4:15pm).