"We understand that a set is a collection of objects, but not every collection qualifies as a set, as exemplified by the collection of all sets (consider Russell's Paradox). To define the collection of all sets, we employ the term 'proper class.'
Now, I would like to confirm if my understanding here is accurate: A class is a collection of objects, and these objects can be sets. Classes are categorized into two groups, namely small classes, known as sets, and large classes, referred to as proper classes. Is this correct?"
Your understanding is correct, and this is how usual set theorists think of classes. However, its formal details can vary by set theories.
In first-order set theories like $\mathsf{ZFC}$, every object is a set. This means first-order set theory technically there are no "large classes." But we see many chances to treat "large classes" that are not sets, like the class of all ordinals or the class of all cardinals. Although large classes do not exist, considering a large collection of sets is still convenient to formulate an informal argument.
Then what are classes in first-order set theories? Let us recall what you may learn from discrete mathematics or logic class that we can "equate" a formula $\phi(x)$ with the collection $$\{x \mid \phi(x)\}.$$ To formally handle large classes over first-order set theories, we do the above procedure reversely. That is, large classes $\{x\mid \phi(x)\}$ are, formally, (first-order) formulas. Thus, for example, if $\mathsf{Ord}$ is a class of all ordinals, the statement $$x\in\mathsf{Ord}$$ is a syntactic sugar of the statement "$x$ is an ordinal." (Namely, $x$ is a transitive set such that $(x,\in)$ is a well-order.)
However, second-order set theories like Gödel-Bernays set theory $\mathsf{GB}$ have classes as objects. You may handle classes as valid objects over second-order set theories, unlike first-order set theories on which we need to exploit first-order formulas to express large classes.
These two approaches are not too different in the sense that every first-order set theory $T$ can be extended to a second-order set theory $T'$, and both $T$ and $T'$ prove the same first-order statements. For example, it is known that $\mathsf{GBc}$, $\mathsf{GB}$ with Choice, and $\mathsf{ZFC}$ proves the same first-order statements.