Are you fat or heavily built? If you’re fat then lose some weight and the climbs will get easier. If you’re heavily built then you might just want to ride uphill more often 🙂
What kind of weights do you do? Big, muscular, and strong doesn’t necessarily translate to strong on the bike, as all the skinny types in the peleton prove. Maybe focus your gym work on leg strength more – squats, lunges, etc. Use weight but do more reps. Also work on your cardio if you’re blowing on climbs – find a good spinning class if you like to go to the gym.
And then of course technique – everyones different but generally spinning is better, with spds so you have a more efficient pedal stroke.
Apologies if this seems condescending or rude, not my intention!
Of course you could always upgrade your bike 🙂