How do run a 120Vac water pump using Arduino starter kit supplies
The problem with the relays isn't that of current, but of voltage. You will need to supply 12V to the coil of those relays in order to get it to energize.

One option is to find relays that only require either 5V or 9V instead, so that it can be either powered from the Arduino's supply or from a 9V battery. You will still need the transistor if using 5V relays, since at that point the coil will need more current than the Arduino output driver can provide.

Another option is to draw power from the incoming AC line and put it through a regulator to get the 12V required to operate the relay. Additionally, a 5V regulator will get you the power you need for the Arduino so that the entire apparatus is powered off the line.