You asked a few questions in your last paragraph. Let's take a step back before answering them, to understand the context.
Essentially, a credit score is an number that's used to predict a consumer's chance of default within the near future. As such, it's used by financial institutions (and sometimes other entities, i.e. employers, service providers, etc) as one of many factors that shape your relationship with them - for instance, whether or not you'll get a loan, but also the interest rate on the loan and potentially the features they require on the loan (i.e. if you have a bad credit score and you're applying for a mortgage, you may be required to carry personal mortgage insurance. Or if you're applying for a credit card, you may be required to secure it with a cash deposit.)
So: Before anyone can answer, for you, if a credit score is important or not, you need to have an idea for how you plan on using credit in the future. If you expect to need a mortgage, or a car loan, it can be helpful to have a good score at the time when you apply for the credit. Of course, it can be hard to plan or predict this 100% accurately, so in general, it's typical advice to understand how scores work and do your best to maintain a good credit score even if you currently have no plans to use credit.
And while it's easy to google and find advice about how to improve your score, it's important to not loose sight of the context: credit scores are meant to be a reflection of your behaviors. If you have good financial habits and are using credit, then, over time, you will have a good credit score, naturally. There will be games you can play to increase it, but in the end, if you're going to participate in the credit economy, it's more meaningful (both for you and for the creditors) if you focus on having good habits versus focusing on getting a certain score.
The good news is, it doesn't inherently cost you anything to keep track of your credit score and make sure it stays reasonably high - and there's really no reason not to do that, even if your goal is to pay cash for "everything" and never use credit. In other words, it doesn't hurt to have a good score, while having a bad score can definitely hurt. Unfortunately, some people take a "just pay cash for everything and it won't matter" approach, which to me, seems a little disingenuous - not everyone wants to pay cash for everything or plan every life situation ahead of time, and glossing over how credit scores work and their importance if you want credit by telling everyone to pay for things in cash misses the point for those people.
To illustrate an example, imagine you're applying for a mortgage The lender will consider many factors in approving you - credit score will be one of them, but typically in terms of the yes/no approval decision, credit score will be less important than income, total debt, and other factors. The credit score requirement to get the mortgage will usually be low, such as 620. So, on the surface, it may not seem important to have a really good score, and it may seem reasonable to dismiss their importance.
However, if your score is in the low 600's versus the mid or upper 700's, you may see significant cost differences - you may be required to carry a PMI policy, and you may see an interest rate difference. If you're buying a $250,000 house on a 30 year mortgage, a $100 a month PMI policy will cost you $36,000 over the life of the mortgage, and a quarter point difference (say, 4% vs 4.25%) in interest rate will cost you roughly $13,000 over the life of a mortgage. That's a difference of nearly $50,000 just because of your credit score. That's nontrivial. And it's something you could have easily avoided with some straightforward behavior changes in the months/years leading up to your mortgage application. Clearly, if you're planning on getting a mortgage, it makes sense to work on getting a high score before you apply.
Which brings us back to your questions:
My question is, how heavily should I prioritize building credit for post-graduation life?
To sum up the above: don't get completely hung up on your score, but do consider your future plans on utilizing credit to decide how much emphasis you want to place on building credit now. If you plan on needing credit within the next few years, it can make a significant impact to focus on your score now, especially since your credit history is short. If you generally have good financial behaviors and you are sure you won't need any credit any time soon, it might not matter as much to really force your score up, but it may still make sense to keep an eye on your score and start building history now anyways.
Is getting a credit card a wise financial decision?
Getting a credit card is not the important decision, really - what you do with it is important. In your situation, where the worst thing about your credit score is that you have a short history, having a credit card is a typical way that people work towards establishing a longer credit history, mainly because cards are a convenient credit tool that's not inherently tied to any specific purpose and which has no inherent cost since you can pay it off and avoid interest completely.
You can get, and use, a credit card without changing your lifestyle or budget, and it can be a great tool not only for improving your credit score, but also for your overall financial health - for instance, if you get and use a card with a good rewards program, putting all your regular expenses through it can net you some extra income.
Are there any other options for building credit that would be more preferable to someone in my situation?
Not really - which is why credit cards are a popular tool. Get the credit card, use it for purchases you've already budgeted for (not impulse buys!), and pay it off immediately. Your score will creep up, you'll be in a better position if or when you need credit.
