Of course, you can just consider the case when $p$ and $q$ are primes. A good reference for your question is, I think, Schoof's monograph on Catalan's equation. The case $q = 2$ is solved in Ch. II, and it involves some arithmetic in the ring of Gaussian integers (the argument goes back to V.A. Lebesgue). The case $p = 2$ is discussed in Chs III (for $q \ge 5$) and IV (for $q = 3$): This is just a little bit more laborious than the case $q = 2$, but still "elementary". In the appendix, you can also find Euler's original proof for $(p,q) = (2,3)$, which is essentially based on the same descent argument, though in a disguised form, used to prove that the group of rational points on the elliptic curve $x^2-y^3=1$ is finite and has order $6$. Some others of your cases follow at once from Catalan's theorem (Ch. VI), whose proof in the book relies on Runge's method (Ch. V).

Of course, you can just consider the case when $p$ and $q$ are primes. A good reference for your question is, I think, Schoof's monograph on Catalan's equation. The case $q = 2$ is solved in Ch. II, and it involves some arithmetic in the ring of Gaussian integers (the argument goes back to V.A. Lebesgue). The case $p = 2$ is discussed in Chs III, for $q \ge 5$ (basic algebra in the ring $\mathbb Z[\sqrt{y}]$), and IV, for $q = 3$ (some arithemtic in the ring $\mathbb Z[\sqrt[3]{2}]$): This is just a little bit more laborious than the case $q = 2$, but still rather "elementary". In the appendix, you can also find Euler's original proof for $(p,q) = (2,3)$, which is essentially based on the same descent argument, though in a disguised form, used to prove that the group of rational points on the elliptic curve $x^2-y^3=1$ is finite and has order $6$. Some others of your cases follow at once from Cassel's theorem (Ch. VI), whose proof in the book relies on Runge's method (Ch. V).