Singlespeed has definitely improve my climbing, as mentioned above its very easy to run through the gears and end up crawling in first gear on a relatively gentle hill.
Coming from singlespeed, I think I would aim to stay in the middle ring yet using the full range of the cassette, and just treat the granny as a bail out gear for proper steep climbing or towards the end of the ride when you are struggling.
Oh, and in the granny, just use the lowest four or five cogs on the cassette (the biggest), middle ring you can use the whole cassette, big ring just use the highest four or five cogs. This avoids running with a bad chain line, excessive wear on components, and risk of chain snappage. To begin with, it would be a good habit to take a look every now and then to check you are not running with the chain crossed up.