Summarize:

The empty roll should be on a plastic spindle or roller. Remove both the empty roll and the roller holding it. Slide the spindle out of the grooves it is resting in, by lifting it upward and to the rear of the dispenser. Smaller tape dispensers may not have a spindle. You should see a plastic core (or roller) with the empty cardboard tape roll surrounding it; remove both of these from the dispenser. Slide the empty tape roll off its spindle, and discard it. If the empty roll is cardboard, it can probably go in with your paper recycling. Remove a new, full roll of tape from the box. Check the box to make sure you've got the correct size for your dispenser. Hold the full roll so that when you pull the tape, the strip of tape will unwind across the top of the roll. The sticky side of the tape will be facing down, toward the floor. (If you are holding it the wrong way, the tape will extend away from you, as it pulls from the underside of the roll; the sticky side of the tape strip will be facing upwards, toward the ceiling.) Slide new roll of tape onto the tape dispenser spindle from either direction; most rollers will fit into the dispenser either way. Hold the spindle so that the paper tab on the end of the roll is facing up and toward the front of the dispenser. Slide the spindle and the new full roll of tape back into dispenser. Settle it into its holding grooves so that it moves freely. Hold onto the paper tab at the end of the tape, and pull it towards the cutting edge at the end of the dispenser. Pull the tape over the sharp edge, and pull down on the tape to cut off the colored tab.
Remove the empty roll. Remove the empty tape roll. Ready a new roll. Align the tape. Slide the tape roll onto the spindle. Align the spindle. Load the tape.