Let $X$ be a topological space. $X$ is called Wedge of circles if $\exists \hspace{0.1cm} \left\lbrace S_{\alpha} \right\rbrace _{\alpha \in S}$ such that :
(i) $S_{\alpha} \subset X \hspace{0.2cm} \forall \hspace{0.1cm} \alpha, X = \bigcup\limits_{\alpha \in S} S_{\alpha}$, $\exists p \in X \hspace{0.2cm} S_{\alpha} \cap S_{\beta} = \left\lbrace p \right\rbrace \hspace{0.2cm} \forall \hspace{0.1cm} \alpha \ne \beta$.
(ii) $S_{\alpha} \cong S^{1} \hspace{0.2cm} \forall \hspace{0.1cm} \alpha$.
(iii) $U \subset X$ is open if and only if $U \cap S_{\alpha}$ is open in $S_{\alpha} \hspace{0.1cm} \forall \alpha$.
With those definition I'd like to prove the following proposition in the case of infinite wedges, since the finite case is represents no problem :
Proposition : $X = \bigvee\limits_{\alpha \in I} S_{\alpha}$ , then
(i) $\pi_{1}(X,p)$ is a free group.
(ii) If $[\gamma_{\alpha}]$ is a generator of $\pi_{1}(S_{\alpha},p)$ then $\left\lbrace [\gamma_{\alpha}] : \alpha \in I \right\rbrace$ is a system of free generators, meaning $\pi_{1}(X,p) \cong F(\left\lbrace [\gamma_{\alpha}] : \alpha \in I \right\rbrace)$
Any help or reference would be appreciated.
What you need is the "infinite" version of Van Kampen. One place which this is stated is in Hatcher's Algebraic Topology, Theorem $1.20$.
It essentially says that, for $X$ a union of path connected $A_{\alpha}$, $\pi_1(X,p)$ is isomorphic to the free product of the fundamental groups of the $A_{\alpha}$ quotiented by some relations, where these relations come from the inclusions of the $A_{\alpha}$ into $X$.
For your case, since you have a wedge, it states that $\pi_1(X,p)$ is isomorphic to the free product of the $\pi_1(S_{\alpha},p)$.
This isomorphism should give you both of the facts you want.