Bike Verbier is the obvious answer, book a guided week with them and get so much more than accomodation, food and drink. They are based in the valley. The headline price may look a little high but Switzerland is expensive and if you add in all you get it’s a bargain, they know the area (not just Verbier) very well and have van uplift options.
Verbier is an apartment and chalet resort so there are very few hotels. If you are on a budget then a small hotel in the valley is your best bet (eg in Les Chables). I suggest you try and rent a small apartment in the village, try tourist office and/or estate agents – you can do that in April. I stayed once in Lourtier at Hotel la Vallee (www.vallee.ch) recommended by Bike Verbier (they where full) – its at bottom of a few enduro trails but quite a pedal (dh) to Les Chables and it wasn’t that cheap.
I usually stay with friends in the village. Any other questions please ask.
Verbier makes a great two centre holdiay with Chamonix which is about an hour way (TdF went up the road from Martigny towards Chamonix this summer).
Finally Morzine is great but it’s nothing like the “big mountain” feel of Verbier or Chamonix. I would hestitate riding too much on my own as there are places if you fall it’s likely to be very serious and another rider or walker may not come past. Verbier (and Chamonix) are about riding the walking paths and some of these are remote and very exposed 😯