Yet another viewpoint on the Thom class.

Given an $n$-dimensional vector bundle $E\to B$ with connectd base $B$, there is fiber bundle pair $(D^n, S^{n-1}) \to (D(E), S(E))\to B$, where $D(E)$ is the unit disk bundle, and $S(E)$ is the unit sphere bundle. Then one may use the fibration long-exact sequences to show that $\pi_i(D^n,S^{n-1})\to \pi_i(D(E),S(E))$ is an isomorphism. Roughly, there is a long
exact sequence $0=\pi_{i+1}(B,B)\to \pi_i(D^n,S^{n-1})\to \pi_i(D(E),S(E))\to \pi_{i}(B,B)=0$ (one can actually prove
this using long exact sequences of each fibration and of the pairs). Hence by the relative Hurewicz theorem (see Theorem 4.37 of Hatcher), $\pi_i(D(E),S(E))=0$ for $i<n$, and hence $H_i(D(E),S(E))=0$ for $i<n$, and $H_n(D(E),S(E);\mathbb{Z})\cong \pi_n'(D(E),S(E))$, the quotient of $\pi_n(D(E),S(E)) \cong \mathbb{Z}$ by $\gamma x-x$, for $\gamma \in \pi_1(S(E))$, $x\in \pi_n(D(E),S(E))$, for $n\geq 2$. If $n=2$, $\pi_1(S(E))\twoheadrightarrow \pi_1(B)$, and $\pi_1(S(E))=\pi_1(B)$ for $n>2$, so this action will be non-trivial iff the vector bundle is non-orientable. Then $H_n(D(E),S(E))=\mathbb{Z}$ if $E$ is orientable, and $H_n(D(E),S(E))=\mathbb{Z}/2\mathbb{Z}$ if $E$ is non-orientable.

Now use the universal coefficient theorem to get the existence of the Thom class.

For $n=1$, $\pi_1(D^1, S^0)\to \pi_1(D(E),S(E))$ is still an isomorphism, but $\pi_1(D^1, S^0))$ is a 2-pointed set. We have
a map $\pi_1(D(E),S(E)) \to H_1(D(E),S(E);\mathbb{Z}/2\mathbb{Z}$ which is an isomorphism (assuming the base $B$ is connected).

Once one has the Thom class, the Thom isomorphism maybe be proved as a consequence of the relative Leray-Hirsch theorem. The proof given in Appendix 4.D of Hatcher is by induction on skeleta; I don't know of a direct proof.

For $n=1$, $p:S(E)\to B$ is a 2-fold covering space, and
the Gysin sequence may be deduced from the long exact sequence coming from the short exact sequence of chain complexes
$$0 \to C*(B) \overset{p^*}{\to} C^*(S(E)) \overset{p^!}{\to} C^*(B)\to 0,$$
with coefficients in $\mathbb{Z}/2\mathbb{Z}$, where $p^!$ is the transfer map. The $(\mod 2)$-Euler class is the element in $H^1(B;\mathbb{Z}/2\mathbb{Z}) = Hom(\pi_1(B),\mathbb{Z}/2\mathbb{Z})$ defining the 2-fold cover $S(E)\to B$.