How would I go about determining if these subsets are subspaces of a vector space? What would be the correct method/approach to evaluate if these two subsets are indeed subspaces of a vector space?

a) $S{1}=\{x\in\mathbb{R}^{29}| x_1+2x_2+3x_3+...+29x_{29}=0\} $

b) $S{2}=\{p\in P_{29}| p(0)=p´(0)=0\}$

For the first expression, I'm thinking about vector x as being linearly independent and solved by the trivial solution x = 0. Then it should have the same or lower dimension than $S_1$ in $\mathbb{R}^{29}$, thus being a subspace. Am I correct?

For the second expression(/equation?) I have no idea how to interpret how e.g. 
$p(0)$ or the derivative $p´(0)$ is related to "$P_{29}$".
I don't understand your argument for the first question (i.e. linguistic comprehension). You should check the following:

$$\forall x_1, x_2 \in S_1, \forall a, b \in \mathbb{R}: a x_1 + b x_2 \in S_1$$

i.e. every linear combination of elements of $S_1$ again belongs to $S_1$. While it is true that $0$ needs to be an element of $S_1$ in order for it to be a subspace, this is by no means sufficient, and by checking the condition I gave, this will automatically be fulfilled.

For the second, do you know what $P_{29}$ is? And if so, do you see why it is a vector space?
For (a), try to find wich linear fonction S1 is the Kernel. ( I assume you know that if F is a linear fonction, then Ker(f) is a subspaces of a vector space.)

What is $P_{29}$?