I am trying to wire in a 3 way switch to my basement landing light that was originally controlled by a 2 way switch. That part I understand the problem I am having is that for my wires that go to the light I have a black and a red wire. My lights in the kitchen and living room are also controlled off that same circuit. Is there any way to wire up the switch and also have constant power going past to allow for the kitchen and living room lights to work off their own current switches?

Three way switches always work in pairs. You can't have just one. There are several ways to wire a set of 3-way switches. Just how you physically wire them depends on where the power comes to, the light or the switch and route that is the most economical in wire and labor. No matter how they are wired, you must always have a 3-wire cable between the two switches. If you have an existing single pole switch it can be replaced with a set of 3-way switches provided you can get a 3-wire cable between the switches. The installation of a set of 3-way switches would not affect anything else on the circuit.

You said you have a black and a red wire connected to the switch. Are there any other wires in the box?