Here are two different definitions of the Hopf algebra structure. As you write it is easier work in the ring $\Lambda$ of symmetric functions in infinitely many variables but it should not be a problem working only in finitely many variables. This would correspond to looking at generators only up to a certain degree, but the expressions below only use elements of lower degree anyway.

From the point of view of the representation theory of the symmetric group, the product in $\Lambda$ can be defined as $$ V \cdot W = \mathrm{Ind}_{S_n \times S_k}^{S_{n+k}}V \otimes W$$ for $V$ a representation of $S_n$ and $W$ a representation of $S_k$; this product is then extended bilinearly. The coproduct then has a natural dual definition: $$ \Delta(V) = \sum_{i+j = n} \mathrm{Res}^{S_n}_{S_i \times S_j} V, $$ where a representation of $S_i \times S_j$ defines an element in $\Lambda \otimes \Lambda$ in the natural way.

A more direct definition is in terms of power sums. Define a coproduct via $$ \Delta(p_{i_1}\cdots p_{i_n}) = \sum_{k=0}^n p_{i_1}\cdots p_{i_k} \otimes p_{i_{k+1}}\cdots p_{i_{n}}. $$ In particular the power sums $p_n$ are primitive elements for this coproduct and they span the module of primitive elements. The elementary and homogeneous symmetric functions are divided powers for this Hopf algebra structure.

The antipode is uniquely determined by the coproduct. You prove this by induction over degree: when you expand $\Delta(x)$, you find terms of lower degree (where the antipode is known) and two terms $x \otimes 1 + 1 \otimes x$, on which you can deduce how the antipode acts. This holds in any graded connected Hopf algebra.

Here are two different definitions of the Hopf algebra structure. One needs to work in infinitely many variables as you indicate.

From the point of view of the representation theory of the symmetric group, the product in $\Lambda$ can be defined as $$ V \cdot W = \mathrm{Ind}_{S_n \times S_k}^{S_{n+k}}V \otimes W$$ for $V$ a representation of $S_n$ and $W$ a representation of $S_k$; this product is then extended bilinearly. The coproduct then has a natural dual definition: $$ \Delta(V) = \sum_{i+j = n} \mathrm{Res}^{S_n}_{S_i \times S_j} V, $$ where a representation of $S_i \times S_j$ defines an element in $\Lambda \otimes \Lambda$ in the natural way.

The connection between symmetric functions and representations of $S_n$ is as follows. The graded piece $\Lambda^n$ is isomorphic to the ring of virtual representations of $S_n$ via the so called characteristic map. A virtual representation $V$ is mapped to the symmetric function $$ \mathrm{ch}(V) = \frac 1 {n!} \sum_{\sigma \in S_n} \mathrm{Tr}\left(\sigma \mid V\right) \psi(\sigma) $$ where $$ \psi(\sigma) = \prod_{(i_1\cdots i_k) \text{ a cycle in } \sigma} p_k. $$ This is in fact an isometry relative to the usual inner product on symmetric functions, and the natural inner product on representations for which irreducible representations form an orthonormal basis. The representation associated to the Young diagram $\lambda$ corresponds to the Schur function $s_\lambda$, so equivalently $$\langle s_\lambda, s_\mu \rangle = \delta_{\lambda \mu}.$$

A more direct definition of the coproduct is in terms of power sums. Define a coproduct via $$ \Delta(p_{i_1}\cdots p_{i_n}) = \sum_{k=0}^n p_{i_1}\cdots p_{i_k} \otimes p_{i_{k+1}}\cdots p_{i_{n}}. $$ In particular the power sums $p_n$ are primitive elements for this coproduct and they span the module of primitive elements. The elementary and homogeneous symmetric functions are divided powers for this Hopf algebra structure.

The antipode is uniquely determined by the coproduct. You prove this by induction over degree: when you expand $\Delta(x)$, you find terms of lower degree (where the antipode is known) and two terms $x \otimes 1 + 1 \otimes x$, on which you can deduce how the antipode acts. This holds in any graded connected Hopf algebra.