It is absolutely untrue that public transportation is "almost non existence", and anyone who says so isn't paying attention very well.

As qatari said, the taxi situation has improved since the Asian games came to an end. Not only are there less people using taxis, but they've put several hundred more taxis on the road since then. Calling a taxi usually requires a 2-5 hour lead time before they can fit you into the schedule, but it's cheap and they are everywhere. Especially at City Center...dozens of them line up in the front waiting for people. Be sure to carry a map though, as I've had a couple drivers who didn't quite know the city that well. The pull out map from the Marhaba guide works nicely.

I've never used the bus service, so I can't really comment on it, but the taxi service is as good as can be expected for Doha.