Actually, the main trick is that $ (-3 | p) = (p | 3) $ and depends on the fact that $ 3 \equiv 3 \pmod 4. $ Quite a time saver. I use the horizontal typesetting of the symbol, which was introduced by L. E. Dickson. The vertical style always makes me think of fractions.
So, why? If $p \equiv 1 \pmod 4,$ then $ (-3|p) = (-1|p ) \cdot (3 |p) = 1 \cdot (p|3) = (p|3). $
Switching to another letter, if prime $q \equiv 3 \pmod 4,$ then $ (-3|q) = (-1|q ) \cdot (3 |q) = -1 \cdot -(q|3) = (q|3). $
What does a value of 1 tell us? if $(-24 | p) = (-6|p)=1$ for a prime $p \neq 2,3,$ we get either an expression $ p = u^2 + 6 v^2, \; \mbox{as in} \; \; \{ 7, 31, 73, 79, 97, 103, 127, \ldots \}, $ all of which are $\equiv 1 \; \mbox{or} \; 7 \pmod {24},$ or $ p = 2 x^2 + 3 y^2, \; \mbox{as in} \; \; \{ 5,11,29,53,59,83,101,107,131,149, \ldots \}, $ all of which are $\equiv 5 \; \mbox{or} \; 11 \pmod {24}.$ As I said, the primes $2,3$ are to be considered separately.
The same thing works with the Jacobi symbol, which is just a product of Legendre symbols. Just an example, $ (-35 | p) = (p | 35).$ Note the required $35 \equiv 3 \pmod 4.$