Why are potassium channels slower than sodium channels?
Short answer
The activation kinetics of Na+ channels are faster than K+ channels. 

Background
Voltage-dependent channel gating basically occurs through three possible states of the channel: open, closed and inactivated (Fig. 1). 


Fig. 1. Gating of a voltage-activated sodium channel. Source: Balseiro Institute.

Basically, ion channels are protein pores in the membrane. Upon depolarization, the pore opens, which is a process with fast kinetics in Na+ channels and slower kinetics in K+ channels (Lacroix et al, 2013). K+ channel activation is in the order of 6 times slower than Na+ channel activation. The slower inactivation of K+ channels allows sufficient Na+ to flow in the cell for the depolarization phase of the action potential to develop (Fig. 2). 


Fig. 2. Action potentials and underlying Na+ and K+ currents. Source: Dundee Med Student Notes.

After opening, the channel is inactivated, which can occurs through a physical "protein-plug" as shown in Fig. 1, or through conformational changes in the pore. This inactivation again is fast in Na+ channels leading to a transient spike in the action potential, and deactivation is slower for K+ channels (K+ channels do not inactivate, but rather deactivate). This inactivation step results in the refractory period of a neuron (silent period after firing). The channel is then de-inactivated and converted to the closed state, after which the channel is again ready to participate in another round of firing.     

Note that neurotransmitters do not activate voltage-gated channels directly. Neurotransmitters can activate ion channels, but those are ligand-gated ion channels, such as the nicotinergic acetylcholine receptor, AMPA receptor or NMDA receptor (Purves et al., 2001).  

References
- Lacroix et al, Neuron (2013); 79(4): 651-7
- Purves et al., Neuroscience, 2nd ed. Sunderland (MA): Sinauer Associates (2001)