I've built apartment blocks and condominiums all over the island, the radio has declared my people filthy rich, and I have so few uneducated workers that my mines and farms only have 1 or 2 workers each. But still my people insist on living in shacks that I regularly demolish to get them to move to apartments. What am I doing wrong?
Are you sure your people are filthy rich? The treasury being full means the government is rich- not the people.
Demolishing shacks is almost never productive. People aren't refusing to move out of spite or laziness- they have nowhere else they can afford to live. 9 times out of 10, they'll just build another shack.
Tropicans will only ever pay 1/3 of their salary in rent (married couples use their joint income). This means, for example, if someone only earns $5, they will not consider renting a house/apartment that costs $2 or more. If you leave the rent of higher class accommodation and wages on the default settings, very few people will be able to afford it.
Make sure you have some housing available for a low cost. A tenement or two with rent set at $1 may not look pretty but it's cheap, they'll house your poor tropicans and it's still better than a shack. Alternatively, if you can afford it, either slash the rents of your apartments or issue the free housing edict.
You should also consider the social security edict. Without it, students & retirees have no income- if you have neither this nor some free housing, they will be forced into shack.