I can't see any disadvantages to you obtaining at least one or two credit cards, if nothing more than to enhance your credit profile for future use (such as a mortgage or car loan).
What you want to avoid is taking on too much debt. Use your card(s) wisely, pay the balance off whenever you can, and don't treat it like "free money", as many people seem to do. Trey to keep your usage of your cards to no more than 20% of the available limit, because anything higher than that makes it seem like you're not making enough of an income to cover your expenses, so you're using credit cards to make up the deficit. The longer you have your cards, the more they improve your credit score, because the age of accounts is a big factor in the scoring models.
As a side note, given what you have in savings, it might be worthwhile looking at paying down some of your student loan debt, depending on the interest rate you're paying. The savings in interest will make up for it sooner than you might think if your rates are typical of what most students are paying. The goal is to help lower your debt-to-income ratio, which is another big factor in scoring your credit.
Something else to keep in mind is that the lower your credit score is, the more you pay for things that might surprise you. For instance, your insurance rates (car, health, etc.) are frequently affected by your credit score to one degree or another (nobody knows the exact amount of this differential), so the better your credit, the better your rates on other things.
I would strongly advise you to get into a better habit of remembering your bills and making sure they're paid on time. I have come across many people who thought they could count on auto-pay, only to find to their sorrow when a payment is missed that isn't foolproof. Start keeping your due dates in a day planner, even if they're set for auto-pay, and check on them the day after the auto-pay was supposed to run just to be sure. Set the payment dates at least 5 days prior to the bill's due date just to give yourself enough time to act without penalty just in case something does go wrong.
As for why your credit score fluctuates, everyone's does. Mine changes week by week depending on how much I'm using my credit cards and as old inquiries go stale and drop off the report. I wouldn't worry too much about that unless the change is pretty significant (more than about 20 points might be a cause for concern). I'd strongly suggest subscribing to one of the free services that tracks your credit scores so you can keep an eye on them, and get into the habit of monitoring your credit just in case you become the victim of identity theft. Many people are victimized and don't know it for a while because they don't take this one simple step.
I hope this helps.