Can I connect a servo directly to arduino uno without a transistor?
The Arduino's GPIO pins can drive the signal of most model servos directly, without any extra transistors.

Driving a model servo signal input is not the main problem with driving servos from an Arduino. 

The main problem is trying to use the Arduino's power supply to power the model servo. With many model servo's, the amount of current the model servo needs to start, or when it is almost stopped is so big that the Arduino's power supply can't maintain 5V. As the Arduino's power supply voltage drops, the Arduino often gets reset, and the program restarts from setup.