Its almost certainly the alternator belt slipping, when you first start the car the load on the alternator is very heavy as it recharges the battery, as the voltage rises the current drops thus lightening the load. Increasing the belt tension if possible should cure it, if this is not possible a new belt will be needed.

A slipping belt would produce a continuous screech, especially when accelerating.

A rhythmic squeaking in time with engine revolutions would be more likely to be, as you say, a bearing in a pulley, the alternator or (my favourite) the water pump.