Answer first: the receptor is a class B G-protein-coupled receptor signalling mainly through Gs and cyclic AMP, and almost every downstream effect people ask about traces back to where that receptor is expressed rather than to what it does when activated.
Receptor density and downstream coupling differ between tissues, so the dose-response curves for glycaemia, weight and nausea are not the same curve. That is the pharmacological basis for titration.
Central effects reach the arcuate nucleus and the area postrema, regions with an incomplete blood-brain barrier. That anatomy is why a large peptide can act centrally at all, and it also explains the nausea, since the area postrema is the chemoreceptor trigger zone.
The incretin effect itself was established by comparing the insulin response to oral and intravenous glucose loads matched for plasma glucose; the difference is what the gut hormones contribute.
Nothing here is medical advice; this is pharmacology.
Tissue distribution first, then signalling. Nearly every question in this tag resolves at the first step.