Pesky hills

I have taken up cycling to improve my fitness and the weight is coming off well having lost 5 stone since February. My fitness has increased so I’m regularly riding 30-35 miles occasionally 40. However my routes are predominantly flat with only slight hills or long gentle inclines. I can do long gentle climbs but on anything remotely steep I just run out of steam really quickly and quite often end up pushing the bike uphill.
I don’t think it’s a gearing issue as I have a giant talon 2 which is quite low geared but it is still dragging 25 stone uphill, is this something that will naturally improve as I gain more fitness or is it something I can improve by specific training?