If you wanna stick to trains it is doable but a bit of hassle, a one way shuttle service from Geneve Cointrin to Chamonix would cost 80/100Chf approx pp. Otherwise, if you can get to Geneve Eaux-Vives train station (near the center of town) you can get to Saint Gervais and then on to Chamonix, takes about 2hours. You can get solely by train from Geneve to Chamonix but it's alot longer involving Geneva to Bellegarde (which is in the wrong direction), Bellegarde to Annemasse or Roche Sur Foron, then on to Saint Gervais and change for chamonix. Instead I'd recommend getting across Geneve to the Eaux Vives station. It's quicker.
Geneva is easy to get around, train from airport to cornarvin central station is 5mins, from there 15mins on tram to Eaux Vives station and from there you can find direct trains to Saint Gervais or if your timing is off a train to Annemasse (regular trains) to make a connection to Saint Gervais. I don't know what your rail passes are - there are full on eurorail passes that would include all your rail travel except maybe the train between Saint Gervais and chamonix and then between Chamonix and Martigny. For prices I'm not sure but you can check on http://www.ter-sncf.com/Regions/rhone_alpes/fr/
I think Saint Gervais to Chamonix is just under 10Euro. Martigny is about 20Euro from Chamonix. From Martigny the train on to Tasch and then Zermatt is no problem.
In Chamonix the camping Nattfodd has linked to is my recommendation too. Chamonix sud has everything on it's doorstep, public transport not necessary unless you want to go to Grands Montets or somewhere further down the valley. If you're dirtbaggin it then try to avoid staying in Geneva unless your flight gets in at an awkward time .....it's a nice place but it's bloody expensive for most things except public transport (and I know a few bars where you can get cheapish beers!).
Depending on the weather, obviously, Mont Blanc is very doable in June. I'll be having a look myself 'round that time. I know a guide who usually starts taking people up from the last week of May on the trois monts route. So the route should be clear and the Col Du Mont Maudit should be easier to get up and down early in the season. Of course that can all change in minutes but you can find good forecasts and descriptions of current conditions at the Maison de la Montagne in chamonix and of course by talking to people who've just come back.