I want to develop an intuition about the concept of an immersion in differential geometry.
Let $\Psi : M \to \mathbb{R}^3$ be an injective, smooth immersion. Set M=(0,1)x(0,1) $\subset \mathbb{R}^2$.
1. Example of $\Psi$, such that $\Psi(M)$ is closed and not contained in any compact subset of $\mathbb{R}^3$:
$\Psi(x,y) := (\frac{1}{x},\frac{1}{y},0)$. Thus $\Psi(M)$ is closed in $\mathbb{R}^3$ and still an unbounded subset.
2. Example of $\Psi$, such that $\Psi(M)$ is not compact, but contained in a compact subset $K \subset \mathbb{R}^3$.
My assumption was to treat $\Psi(M)$ as a subset of $\mathbb{R}^3$ relative to the topology of $\mathbb{R}^3$. Now I'm unsure if this is the right approach.
Your example in 1. is indeed unbounded, but it is not closed: the point $(1,1,0)$ is the limit of the sequence of points $$\left(\frac{n+1}{n},\frac{n+1}{n},0 \right) = \Psi\left(\frac{n}{n+1},\frac{n}{n+1}\right) \in \Psi(M) $$ but $(1,1,0) \not\in \Psi(M)$.
Regarding 2, I would simply suggest that you write down formulas for very simplest example of immersions, and then attempt to use the formula to prove the properties of $\Psi(M)$ that you are asked to prove.