Q: My car moves in all gears but reverse. Is something wrong with the transmission?

yesterday i was driving to work when my car just stopped. i pushed on the gas but my car wouldn't move in any gears. when i would put my car in reverse it would move backwards. but when i would put in Drive or Neutral it wont move forward no matter how much gas i would push. i had my oil changed Oct.18 and put in a new battery. i opened my hood to see if i had any transmission fluid and my transmission fluid was very low but red. it wasnt burnt or dark. so i put transmission fluid in my car. i started my car put it in all gears one at a time and then turned it off again. i checked my transmission fluid again. it was full and red. then i went and started my gar again put in drive wouldnt move. i put it in neutral and it drove. when i was almost home i put it in drive gear and it was driving smoothly. i didnt want to drive in neutral to mess up my transmission. i parked my car thinking everything was ok. until next morning i started my car and put it in reverse and it only moves forward.

My car has 15014 miles.
My car has an automatic transmission.