Is something wrong with my gauge or do I really have a temperature problem?

My temp gauge, which is usually at 11 o'clock, has been going to 12 straight
up and, sometime, even to 13, but never really past that. I put some 50/50
coolant in it and it continued to do the same. Tonight, I drove to work and it
was straight up all the way like usual, but the car seemed fine. Then, as I
parked it and put it in gear to get out, suddenly, I hear a bing and noticed that
in about three seconds, it's fully in the red. I've needed an oil  change for
some time now (I don't know if that has anything to do with it) or if it's a faulty
needle or if there is really something wrong temperature-wise. I need some
guidance, thanks!