I have a random variable $X\sim\mathcal{N}(\mu,\sigma^2)$. Since all moments of a Gaussian are finite, I know that $$\mathbb{E}[|X|^p]<\infty, \ \forall p.$$
What does this tell me about the individual realizations of $X$? Can I say anything like $|X|^p<\infty$ almost-surely?
A single realization is just a real number, and thus always finite.
A finite sample of a gaussian distribution can have any value with finite probablity. Therefore you can not get any "almost-sure" property there.
But you can for increasing number of samples. Such as: The mean of a sample of size $n$ converges almost surely to the true mean $\mu$ for $n\to\infty$. (This is not true for general proability distributions, but it is true for gaussians due to the finite moments.)