If I had the choice to move to an arbitrary region on the planet, which regions should I favour and which should I avoid in terms of habitability in face of the possibility of an extreme climate change?
The places on Earth that have historically had the most stable climates are typically within the tropics and close to sea level/coastlines. The tropics are not influenced as much by the rest of the Earth's weather patterns since it is more of a source of weather (e.g. the strong convective atmospheric motion that moves air away from the tropics due to the dominant solar heating). Staying close to coastlines ensures that any rising global temperatures are moderated by the abundance of water. Staying close to sea level ensures that you are not influenced much by changes in glacier activity nor snow-melt elevation changes.
There is literature that states that tropical ecosystems are most sensitive to climate change, and that tropical areas will see climate change sooner. However, my point is that the climate itself will be the least extreme in the tropical regions near the ocean. Furthermore, I would like to point out that climate change per se will not make most of the planet uninhabitable as the question insinuates. As the climate changes, there will certainly be very large changes to the land-surface and ecosystems... including desertification of some areas. However, any regions with sustainable ecosystems will be habitable, given that there is a source of fresh water large enough for the human population (which the Tropics often have, due to frequent rain... though areas dependent on snow-melt feeding rivers will suffer).
Places with stable climates like interior deserts are likely to be least affected by climate change. They will go from being simply "hellish" to moderately infernal. In Middle Eastern deserts for example, summertime temperature extremes will only rise perhaps another 5-10 degrees F, but will be enough to be mostly unsurvivable. Climatological fringes like equatorial mountain ranges, and polar latitudes will suffer the most, as they are currently.