You can climb a hill basically three different ways:
- Spinning up the hill in low gear at high RPM
- Charging up the hill standing up (or at least really pumping the muscles sitting)
- Grinding up the hill at 20-30 RPM
Of those, the third one is a good way to wreck your knees and generally make yourself miserable. The other two, however, are both perfectly acceptable.
Basically, you divide hills into those that are short enough you can comfortably "charge" up, and those that are longer and more challenging.
For the short ones (and "short" is purely a personal choice) you typically downshift a bit and then raise your exertion level to something near your comfortable maximum, probably dropping your RPM a little but maintaining at least 60 RPM.
For the longer hills (and maybe a few shorter ones at the end of a long day) you downshift a fair bit more, raise your exertion level only a little, and "spin" at roughly the same RPM you'd use on level ground (though truth be told mine always falls off a bit), adjusting gears up/down to maintain that RPM and exertion level.
The thing you want to do is to use your gears to maintain the "sweet spot" where you're holding a relatively steady RPM (somewhere between 60 and 90) and also maintaining an exertion level that you can sustain for the long haul.
Of course, if your bike doesn't have a broad enough gear range to "spin" up longer hills then you have a problem. You can try "charging", but that can run you ragged pretty quickly. So (since you should NOT "grind" up the hill) you probably need to figure a different route that is more suitable to your bike and fitness level.
One basic rule of thumb I use while cycling that seems to work out pretty well is to always pedal faster than I breathe. That is, if my respiration rate is 60 then I should be pedaling at least 70-80 RPM. If I notice my RPM falling off or my respiration rate climbing, I downshift until my respiration rate drops a bit. And on the flat I aim for an RPM that's twice my respiration rate.