People often incorrectly/wrongly believe they will get more pleasure from having more money, but in actual fact they don't. A lot of research suggest that the things money can buy don't make people happy. If given the chance, most people soon learn that they won't find the good life they expect to buy/buying expensive things.
To begin with, do both incorrectly and wrongly convey the same idea?
Second, does in actual fact read good?
Third, would you use to buy or buying in the above context? Thanks.