This type of question often rises when starting to learn the theoretic aspect of computer science.
The answer, which may seem unfortunate at first, is that there is no algorithmic way of determining whether a language is regular. Moreover, there will never be an algorithmic way, no matter what advances in computers the future may bring, and how many clever people try to come up with algorithms.
Now, this is a pretty strong claim, so I should back this up. Technically, the reason for this is that the problem of determining, given a Turing Machine, whether its language is regular, is undecidable.
Presumably, since you're just getting into automata theory, you're not familiar with undecidability yet. But when you will be, this will all make sense. Also, it will all be very very exciting, as these results are simply mind blowing (in my opinion).
As an additional note, the question "is a given language regular?" should be carefully formulated. Specifically, how is the language given?
- If it's given by an automaton -- then it's regular (but that's just silly).
- If it's given by a regular expression -- then it's regular (but that requires some work to prove).
- If it's given by a Turing Machine -- it's undecidable whether it's regular.
- If it's given by a different model, the it depends. For example, for CFG's, the problem is still undecidable. But for simpler models it may be decidable.
I guess all of this is not what you were aiming at with your question, since you were probably given a certain language described in English, and you're just trying to get intuition on whether it's regular.
The good news on this front, is that by the time you solve 10 such questions, you'll probably have enough intuition for all the exercises you'll be given.
A good rule of thumb is that regular languages cannot "count" arbitrarily. They can count up to a fixed number (e.g., couting modulo 6, counting up to 13, etc.) but they cannot e.g., compare the number of a's to the number of b's.