Should someone that has never had a credit card before check his/her credit score before applying?
It wouldn't hurt.
Even if you haven't had any credit or loan activity, someone may have gotten a card or a loan under your name: identity theft.
That gives you the opportunity to clear things up before you apply the first time.
Granted, it may be a long shot, but as littleadv said: "Why not?"
One thing to keep in mind is that the credit scores you get online are not your actual FICO scores, but your PLUS score, which is some fictional number that is not what lenders see.
However, if you want to know how your credit looks, it's still decent information to get. Go for it. The important thing to really know is your credit report. You always want to know if there is anything negative on your credit report.
If this is your first credit card, I am guessing you have very little or no credit history. That itself is negative. I was denied my first credit card and that is probably why.
If that happens, keep trying until you get one. The only way to get credit history is to have credit and to use it.