What is the difference between a reward and a value for a given state?
Let us say that an agent took an action from state $A$ and reached state $B$ and got a score $R$. This instantaneous score the agent received on reaching state $B$ is called the reward.
Now, let me introduce you to the concept of return. Assume that an agent followed a particular trajectory:
1. State 1 -> Action 1
2. Reward 1, State 2 -> Action 2
3. Reward 2, State 3 -> Action 3
...
n. Reward n-1, State n (Terminated)
Return (often denoted by $G$) is the sum total of all the rewards obtained by starting from a state State 1 and following a policy.
So, the definition of the return is
$$G(s_1) = R_1 + R_2 + R_3 + ... = \sum_{i=1}^{\infty}R_i$$
Sometimes (most often) these sequences never terminate, so we include a discount factor (Greek letter gamma, $\gamma$) to rewards obtained in the future.
The definition of the discounted return $G$ is
$$G(s_1) = R_1 + \gamma R_2 + \gamma^2 R_3 + ... = \sum_{i=1}^{\infty}\gamma^{i-1} R_i $$
$\gamma$ is a number between $0$ and $1$: it defines how much importance the agent gives to long-term rewards. For a smaller value of $\gamma$, more importance is given for short-term rewards.
Now, coming back to your question. A value of the state is the expected return for an agent starting from that state and following a particular policy. In the case of stochastic policies (policies that have inherent randomness) and/or for environments with stochastic transition probabilities and/or stochastic rewards, the value is the sum of (the returns of all trajectories multiplied by the probability of taking that trajectory).
And should the initial values of the states always be set equal to zero?
Not necessary, zero initialization is one of many ways to initialize. Random initialization is another method. It depends on the environment setting.