You want to sightsee and get into another culture ? My hands down suggestion would be to go to India . It has one of the most diverse populations in the world and is the home to many monuments. Every state has something to offer in terms of cultural diversity as well as places to visit. You wouldn't be disappointed . And in fact, since you say money wouldn't be a big concern , you could very well end up visiting most of the monuments in the country within 3 weeks.