You should memorize the order of sharps and order of flats downward (F#, C#, G#, etc) because they are the order of sharps that appear in sharp key signatures. For sharps, you could use the phrase "Fat Cats Go Down Alleys Eating Birds", and for flats, the word BEAD followed by Greatest Common Factor.

For the order of keys, that should be memorized in both directions. Question: Are these orders in any way related, that you can see?

The way the orders of sharps and flats work, is that key signatures always contain the first few sharps, for example, in the key that has three sharps, it's the first, second and third sharp, F#, C# and G#. The same is true of flat key signatures, for example, in the key that has four flats, the flats are: Bb, Eb, Ab and Db.

Finally, when you start reciting the notes, it will save you a thinking step, if you always recite: "eff-SHARP, cee-SHARP, gee-SHARP, etc.