I replaced a receptacle in my kitchen that is 1 of 3 in the circuit. The existing wires are 14/3. I rewire the receptacle black black white white, and twist the red together and only the bottom receptacle works. I check with the other 2 receptacles and only the top one works for them. I also took a look at the middle receptacle and it is using 1 red and 1 black for hot. I'm not sure what original intent was but I would really like all my receptacles to be fully functional.

Any ideas?

The red and black are likely 2 different circuits. You should only have 2 recepticles on each circuit in the kitchen.
The recepticles have a tab that separates the top from the bottom, allows 2 circuits, or maybe half switched.
Tab should be intact if for 1 circuit.
For 2 circuits, break tab connect red to top, and black to bottom.
Should also be #12.