As above really, just keep on at those hills !
I wouldn't get too worried if you've only been cycling 2 months, that's a very short time even tho it seems you are doing very well indeed. I live in west wales where u can't avoid them so I haven't had choice in the matter and even tho I class myself as relatively fit, it's taken me 6-8 months to get used to them. Core exercises will help I'm sure but the best thing you can do is just get out on the bike as much as you can. Otherwise, things can get over technical.
With the whole getting out the saddle thing....iv found that sometimes when encountering a short steep hill I will almost certainly get off my arse and gun it!
But with the long drawn out ones I don't get out the seat all that often, literally just to change position now and again. So I would say the best thing is to keep practicing those hills