Hi
Just got a turbo trainer and joined Zwift. Really enjoying it, using it to try to work through a bout of depression and to build up my physical fitness too.
Happy with my pace on the flat, but would love to be able to climb reasonably proficiently. Is this likely to be down to building cardio fitness & taking on more hilly routes or am I likely to need to start doing some weight training to strengthen thighs and calves?
Sorry if a regular q, total newby to this world 
Welcome to the forum.
I am not a trainer, but I would say if you want to go faster on the hills then you need to ride more hills.
Weight training is never a bad thing but the truth of it is that an hour lifting is an hour you could spend riding. Spend the time working on your fitness. Specifically sustained power 
. Then work on losing any unnecessary weight. Your power to weight ratio is key for sustained climbs.
“Dont buy upgrades, ride up grades” - Eddy Merckx
Thanks for the tips, trying out some climbs and so far so good 
practice doing some sustained efforts out of the saddle without blowing up.
try some of the zwift workout programmes. once you have an accurate FTP, these are invaluable for making gainz.