Power Outlet -3 Wire Conductor on Tandem Breaker Wiring
On the old receptacle were the break-off tabs connecting the contacts on the hot side of the two receptacles removed? What about the break-off tab on the neutral side--was it removed or in place? If you just want to use only one hot (red or black) to power both receptacles of the the duplex receptacle, then on the new receptacle, don't break off either of these connecting tabs.

But if you would want to power the two receptacles independently using one neutral (i.e., a MultiWire Branch Circuit (MWBC) using both red and black and the one white), you must put in a 2-pole breaker and put it in a different spot than this tandem breaker. This might require rearranging the breakers in your panel. If you would try to insert a 2-pole breaker in the same location as this tandem breaker, I believe you would find that the 2-pole would not snap into place. It is that way on my GE panel and I think this is a universal requirement. 

And on the new receptacle you would remove the break-off connecting tab on the hot side, but leave the tab on the neutral side intact. (You have only one neutral and you want it to serve both receptacles.

In any case do not connect the red and black wires together. You either abandon one of the hots and cap it or, if you use a MWBC, you connect each hot to "its own" receptacle.