Transitions of viscous flow between coaxial conical cylinders with the inner one rotating and the outer one at rest were investigated to reveal mode selection of the first instabilities with the aid of flow visualization and spectral analysis. The rotational velocity of the inner conical cylinder was linearly accelerated from rest until reaching its final speed. The different observed states were successfully distinguished by their dependency on the acceleration rate  in the investigated range of Taylor number Ta. Transitions between states were determined as functions of  and Ta by fixing the Taylor number and varying the acceleration rate in the range 0.01-1.5 rad/s². Observed states were classified into: first toroidal vortices (FTV), helical motion (HMV), upward travelling vortices (UTV), steady Taylor vortices (TVF) and wavy vortices (WVF).  Modes of six pairs of Taylor vortices (6TVF), seven pairs (7TVF) and eight pairs (8TVF) were observed at the same Ta and different . Steady Taylor vortices and wavy vortices were also observed when  increased at the same Ta. The spectral analysis indicated that the states HMV and WVF have constant ratios between the characteristic frequencies and the frequency of rotation of the inner conical cylinder, while in UTV the ratio decreases with increasing Ta. The mode selection diagram in the (Ta,  ) plane has no regular form with regard to the zones delimiting the different observed states.