Practically, mathematicians today should know the rudiments of programming in at least one language (Mathematica and MATLAB count). They should know the basics of probability and linear algebra. They should know these three things because if they get jobs outside of academia they will generally be expected to use at least two of these three, and probably all of them.

Mathematicians should know how to use the internet and how to learn there. They need not recall many formulas, as the convenience of having them at one's fingertips can be "outsourced" to the internet. By the same token, they need not even recall most of what they have learned, but instead should be able to refresh their memory quickly.

Classical and complex analysis have clearly (I think) become less important to command in detail. Combinatorics and algebra have become more so. This is because of computers, and the interplay between mathematics and technology more generally.