Let $X(t),Y(t)$ be two independent Poisson process with rate $\lambda_1, \lambda_2$. That is, at any time instance $t$, $P[X(t) = k] = \dfrac{(\lambda_1 t)^k}{k!} e^{-\lambda_1t}$.
I wish to show that $X(t) + Y(t)$ is a Poisson process with rate $\lambda_1 + \lambda_2$
Suppose that $X(t)$ and $Y(t)$ were Poisson random variables, then this problem is usually approached by transforming $X,Y$ into frequency domain by way of characteristic functions. Summation of PMF turns into multiplication of characteristic function and we immediately arrive at the result.
However, does the result still hold when $X,Y$ are stochastic processes? Why or why not?
We know that, for a given $t$, $X(t)$ and $Y(t)$ are independent Poisson RVs with means $\lambda_1t$ and $\lambda_2t.$ So, then, using the Characteristic function approach or otherwise, it follows that for all $t,$ $X(t)+Y(t)$ is Poisson with mean $\lambda_1t +\lambda_2t = (\lambda_1+\lambda_2)t.$ You are right that this is not sufficient to show that it is a Poisson process, although it's certainly a good start. You've shown that the marginal distribution is correct at each $t,$ but not necessarily the joint behavior for different $t$'s.
To prove $Z(t) = X(t)+Y(t)$ is a Poisson process, you would need to prove all the defining properties of a Poisson process hold, according to whatever definition you are using. For instance, one property that is often part of the definition is that for $t_1 < t_2 < t_3 < t_4,$ $Z(t_2)-Z(t_1)$ and $Z(t_4)-Z(t_3)$ are independent Poissons with means $(t_2-t_1)\lambda$ and $(t_4-t_3)\lambda$ where $\lambda$ is the rate of the process. This is a stronger property than just the marginals being right.
It is pretty easy to show here. We have $$ Z(t_2)-Z(t_1) = (X(t_2)-X(t_1)) + (Y(t_2)-Y(t_1))\\Z(t_4)-Z(t_3) = (X(t_4)-X(t_3)) + (Y(t_4)-Y(t_3)).$$ From addition of Poissons and the fact that $X$ and $Y$ are Poisson processes (so they obey the above property), we can immediately see that $Z(t_2)-Z(t_1)$ is Poisson with mean $(\lambda_1+\lambda_2)(t_2-t_1)$ and $Z(t_4)-Z(t_3)$ is Poisson with mean $(\lambda_1+\lambda_2)(t_4-t_3).$ The fact that they are independent from one another follows from the independence of $X(t_2)-X(t_1)$ from $X(t_4)-X(t_3)$ (and the analogous fact for $Y$) as well as the fact that $X(t)$ is independent from $Y(t')$ for all $t,t'.$
This goes a long way toward proving $Z$ is a Poisson process with rate $\lambda_1+\lambda_2.$ There are a couple other things you need to show, and it will vary depending on the definition you're using.