How should I approach hills?
I try to put out a constant amount of effort no matter what slope I'm on:


A constant 'cadence' of 60 to 90 RPM (that's how fast you spin the pedals)
A constant force on the pedals


The useful energy you put in is proportional to a product of the force multiplied by the cadence: spinning faster at the same force results in more energy input.

To keep my cadence and force constant, I change gears: if it's spinning too easily/quickly and I'm going too slowly, then I change up into a higher/harder gear; on a hill if it's too hard to spin (at 60 to 90 RPM) then I change down into a lower gear.

I'm always seated (not standing on the pedals ... except when riding over bumps on the road).

With many (21+) gears you can find a gear (or a range of gears) to match the slope.

The pros on TV are sometimes sprinting etc. You don't necessarily want to ride like they do, when you're commuting (and a self-described newbie).

FWIW now that I'm more practised (5 times/week since March) I sometimes find myself taking a hill in a bigger gear than I used to: but still sitting, and spinning quickly.




  Obviously its easier to pedal in lower gears but then I feel like I'm just wasting my energy pedaling quickly and not getting anywhere.


When approaching a hill from the bottom, approach at a good speed: fast cadence, high power, and a big gear.

When you're onto the up-hill, increase your push slightly (to try to maintain your cadence even though you're now going up-hill). Depending on various things (short hill, enough reserve power) you might thus get all the way up the hill without changing gear (but this might be exhausting) and finish before you're even tired (you complete it quickly because you're going quickly).

If you can't power up-hill in the big gear then your speed (and therefore your cadence) begins to drop. It's not worth letting your cadence drop: if you allow your cadence to halve, for example, then to maintain your power output you would need to double your pedal force to compensate (which you're not strong enough to). So whenever the terrain drags your cadence down to some minimum (e.g. 70 RPM) then shift down to an easier gear to compensate. Don't change down too far: only one gear at a time and only as necessary.

The end result is that you're always pedalling at a good cadence and with a strong force: and that is as fast as you can go! Not "wasting your energy".

When you change gears (especially on a hill) it's convenient if you have nice gears which let you shift quickly while you're loading the pedals, or which require you to ease up on the pedals (to avoid grinding and jumping the drive chain) for only a small fraction of a second.



Postscript:

I since moved to a hilly area (a few years ago) so I've had more practice. Wearing bike shoes (so my feet can't slip from the pedals), I now stand on the pedals to climb hills. My endurance has improved, so now I can sustain that for about 100 breaths consecutively before getting tired and sitting down.

On a horizontal, you want a sustainable effort (you don't want to become exhausted after a couple of minutes).

On a hill you may use a higher, less-sustainable effort because:


If you reach the top of the hill before exhaustion then you won't need to sustain it
If the ride is a mix of faster (flat) and slower (up-hill) segments, it's the slower segments that take more time, and so are those into which it's worth putting any increased effort.


Standing on the pedals puts quite a bit more weight (force) on them, so you can be in a higher gear and go faster etc. I almost never use the bike's lowest gears any more: the hills up which I used to spin, I now climb standing.

In summary I think my earlier advice above is good, but with practice some people grow out of it.