If I want to connect two picaxe chips together, do I need any resistors or anything like what is required to connect to the computer, or do I just need 1 wire (only send) and the same ground for both of them?
Correct -just pin to pin wires (& common ground)-it couldn't be easier. Picaxe serial comms is so robust that data will even travel thru' damp string (or kids holding hands) & ground returns. Stan
In many cases, as Stan says, no resistors needed.
BUT, if the PICAXEs are on different boards with separate power supplies and/or you are using a PICAXE which has settable I/Os then stick in a resistor.
I'd use them in many cases in any event as they a) Will provide peace of mind and b) form a handy link on home-made s/sided pcbs.
Also, of course, if different power supplies were used, and no ground/earth loops were allowed, then consider using opto-couplers. 1 in each direction. 4N25 etc are good for jobs such as this.