>>1343231I'd recommend either joining a good mountaineering club (I don't think many of these exist, they tend to be pretty rubbish, except the ones at universities ofc) or getting some lessons from a guide (expensive, unfortunately). I was lucky enough to learn from friends while at university.
As for where to go, I'd recommend the Ecrins; cheaper than Chamonix, better weather (statistically way fewer rainy days per summer month than Chamonix), fewer lifts which saves you money. Plus lots to do at easy enough grades and there are big bolted climbs to do from the valley as well as bouldering in the valley.
Pic is OP's friend climbing the Migot spur, Aiguille du Chardonnet