From what I know, the truth of an analytic statement is based solely on its meaning so understanding the meaning of an analytic statement is enough to determine whether it's true. I understand why tautologies such as "either it will or it will not rain" or statements that are true by definition like "all bachelors are unmarried men" would be considered analytic statements, but I still struggle to grasp the concept. Are there less obvious examples of analytic statements?
I think a good example of less obvious statements that would be considered analytic are theorems of mathematics - if everything is well-defined, you have a set of axioms, and you follow some given rules of deduction, then the theorems which follow from the axioms are purely analytic.
For example, Euclid's "Elements" is based on some set of axioms and rules of deduction, from which you can analytically derive the Pythagorean theorem - a nontrivial analytic statement.
I think you've largely got the idea down.
Analytic means something that can be shown true without reference to anything else. This means either pure tautologies or expansions of definitions (if there's some other ingenious category, it does not spring to mind).
The key value of analytic is that it is part of a pair. The other feature in the pair is synthetic where we learn something by adding to two different truth things together, such as combining two axioms. As generally used in philosophy, that's how the pair of terms works.
For most of us, we know the terms through Kant where it occurs in tandem with a two further pairs: necessary vs. contingent and a priori vs. a posteriori. (These, like synthetic / analytic, does not truly originate in Kant but these terms are studied the most in trying to understand Kant). Here, the idea is how do I know something. Something that is a prior analytic can be known just by thinking of the term and working out what it means. That which is a prior synthetic involves the conjunction of multiple a priori truths to reach its conclusion.
A posteriori items work based on evidence. These two can either be things that are identities or things that require working together multiple pieces. If my memory is correct, something that has an a posteriori element would then make the entire argument a posteriori for Kant.
A second big name on this topic is Kripke who is famous for proposing the idea of a posteriori necessity in Naming and Necessity
There's some more examples in a different question: What are examples of analytic a posteriori knowledge?