I find that using my primary hand (my writing hand) is better than the other one.
Keep eyes on the road, you should not have to look to find your bottle.
I use my teeth to open the sippy valve, breathe in and then take a small drink and hold it for a sec before swallowing.
Sometimes I hold the bottle over the bars for 5-10 seconds to let the first mouthful have an effect, and then take a second drink.
When done I use my chest or my index finger to close the valve, and then put the bottle back in an empty cage (yes once I tried to push a bottle in where the other already was)
You should be able to put the bottle down into the cage without looking, although sometimes I take a quick glance if it doesn't go straight in.
Drinking on a climb is okay as long as its not too steep. Straight and consistent is best.
If you're riding in a group, don't drink on the front cos you'll slow down and that will cause the whole bunch to loose momentum. Wait till you rotate off the front before drinking. Right at the back is ideal.
Never drink on a downhill or anywhere the road surface is a bit sketchy. With a bottle in hand your control is lower, and gravel or sudden potholes need a lot of attention.
Improve by practising - same as most things in cycling. Quiet ride make sure there are no cars around and nothing coming up that may be an obstacle, and have a sip. Don't guzzle water, it will be bad for your guts and there's a chance of some going down into the lungs by mistake, inducing a cough.
I find that I often phlegm-up within 5 minutes of a big drink, so drink less but more often.