I think that one of the most interesting thing about sequences is that sequences are "easy" to manage.
In a first place, when you modeling something, a recurrence relation often arise. When you have this relation, you can compute (by hand or with a computer) all the terms you want.
But in a lot of cases, your phenomenon is continuous (e.g. physical movement), hence the most appropriate way to model that is (real) functions. However, it is hard to deal with functions $\mathbb R \to \mathbb R$. So you discretize your problem using sequences (e.g. Euler method to solve ODE).
It is the same in a more abstract point of view: when you are doing some topology, the spaces can be very very difficult to manage. For example, the set of the continuous functions over $[0,1]$: here, manipulate sequences of functions is easier than work with big (non countable) bundle of functions.
So we try to turn properties concerning "open sets", "compactness", etc. into properties concerning sequences as much as we can: for example in a metric space, a set $A$ is closed iff every sequence converging converges in $A$ (which is often simpler to use than "$A$ is closed iff its complementary is open").
That's why it is important to be fluent with sequences! If you are not, it will be difficult to understand well more abstracts concepts (since those concepts use sequences a lot).