What kind of gearing do you have on your bike? You need to have low enough gearing that you can pedal up any hill without killing yourself. I rode a fully loaded bike in the Adirondacks and only had to walk a couple of hills (at the end of LONG days in the saddle when I was just pooped). I am also small and lightweight, so should be a strong climber, but I don't have a lot of power -- it's all about power to weight ratio. My touring bike is geared very low -- a triple crankset in front with a 34T in back (i.e., mountain gearing), so there are no hills I can't climb in it.
Some of it is just training; you'll get stronger in time, but appropriate gearing for the riding you do most often is very important too.
Also, technique is important. Make sure to gear down before you hit the bulk of the hill. It can be very difficult to shift under strain as you need to ease off the pedals just a bit when shifting. If you shift under a lot of strain on a steep hill, you can easily lock up your chain -- I've done it. You need to anticipate and shift ahead of time.
I am sure others will chime in with even more good information and details. I was writing this in a rush, as it is bed time. Best of luck, and don't give up!
Emily
2011 Jamis Dakar XC "Toto" - Selle Italia Ldy Gel Flow
2007 Trek Pilot 5.0 WSD "Gloria" - Selle Italia Diva Gel Flow
2004 Bike Friday Petite Pocket Crusoe - Selle Italia Diva Gel Flow