Digital Outputs can each sink and source around 20mA and are capable of driving LEDs and other small loads directly. Be aware that while each output can handle 20mA there are limits for each port and for the entire chip that typically prevents 20mA being used on every output. For some PICAXE chips, the port and total limit is around 95mA while for others the limit can be around 200mA. A review of the datasheet for the core PICmicro chip is recommended -- Wiki
You'll be needing a transistor driver to drive the relay. If the relay pulls less than 150mA you can use a small signal switching unit like a 2n3904, for up to 800mA a metal can 2n2222 will do. For big loads (several amps) perhaps a MOSFET would be the way to go. Always clamp the relay with a rev. biased diode.