What are some mnemonics to help one remember that Injection = One-to-one and Surjection = Onto? The only thing I can think of is 1njection = 1-1.
An injection $A \to B$ maps $A$ into $B$, i.e. it allows you to find a copy of $A$ inside $B$.
A surjection $A \to B$ maps $A$ over $B$, in the sense that the image covers the whole of $B$. The syllable "sur" has latin origin, and means "over" or "above", as for example in the word "surplus" or "survey".
Take a look at this picture (from Wikipedia):
This function is NOT injection, because two arrows point into single point in that picture.
Now imagine injections at the doctor. Injections usually hurt and you, sure as hell, woudln't want anyone to stick that injection into the same point on your body multiple times.
So that's why injective functions cannot have multiple arrows pointing into the same point (value)
The way I remember it is that when you get a flu shot your entire body doesn't turn into a giant flu virus, because the needle is smaller than your arm is. Then you can easily remember surjection as "the other one".
Another one is that in-jections are in-ferior and su-rjections are su-perior.