Investigation of neurotransmission can be a very wide topic to cover here. I suggest you would be more specific to get a good answer. Studies on neurotransmitter systems such as dopaminergic, serotonergic, orexinergic and so on can vary from their concentration, synthesis, kinetics, and chemical characteristics to a more experimental studies such as their release in synaptic cleft, their neuromodulatory roles in emergence of specific behavior or neuronal activity and their dysfunction in neurological disorders.
Techniques include: molecular (immunoblotting for their expression, ESA for their concentration), microscopy and imaging (using tracers and fluorescent antibodies to track their movement and release), electrophysiological (to supplement the molecular data on release and their role to propagate the action potential and neuronal message to another cell), optogenetics (to manipulate their release or target their synthesis), etc.
In continuance of DR. Bahman Sadeghi , i strongly recommend that if its only meant for estimating the concentrations ( be it in Plasma/Urine CSF or in any other body fluids) of Neurotransmitters such as Dopamine, Epinephrine, 5-hydroxyindoleacetic acid, 5-Hydroxytryptophan, 5-Hydroxytryptamine(serotonin) , you should go for the HPLC.