Given a quadratic form over $\mathbb Z$, how do I determine for which primes $p$ (including the infinite prime) I need to use to find a solution in $\mathbb Q_p$ in order to produce a global solution using Hasse-Minkowski? Every summary I have read says that this procedure involves only finite computation (checking primes dividing numerators/denominators of coefficients if the quadratic form is diagonalized), but this is never justified. Why is this true?
Closely related, how do we actually find the solutions for these primes? I have never seen an algorithm or suggestion to compute or determine the existence of real solutions, and for $p$-adic solutions, the best explanation I have come across amounts to "use Hensel to find a solution mod $p$ first (which makes perfect sense), and use Quadratic Reciprocity to find the solution mod $p$." How does the Quadratic Reciprocity step look when there is more than one variable involved?
In summary, which primes do we need to check, and why? What is a sketch of the mod $p$ algorithm using Quadratic Reciprocity? Is there a generic algorithm or algorithms for finding real solutions?
For future readers of this post, I should also ask for sources should one consult to find more detailed answers to these questions and/or for further reading.