See http://www.insectscience.org/8.71/ for more detailed description and reasoning.
Step 1: Pick an Ant Species
You will be utilizing an evacuation escape behavior of ants that are preyed upon by army ants. Not all ant species that are preyed upon by army ants have this response behavior, but several species do. With a little natural history literature research or field observations should be able to find this out.
In this demonstration we are collecting the ant Aphaenogaster cockerelli using the army ant Neivamyrmex nigrescens. Both are common to the southwestern US.
Step 2: Find the Army Ants
In this example, N. nigrescens hunt primarily at night. They form dense trails appearing from seemingly nothing but a small hole in the ground (see video below). The ants we are after respond to army ant raids at night by rapidly evacuating their nests. This behavior can be triggered by releasing army ants at the targeted ants nest entrance.
In this case, army ants are encountered by randomly walking, after dusk, and scouring the desert floor for a raiding column. Once you find the army ants, collect as many (live) as you can! (see next step)
Step 3: Bring a Collection Device
An aspirator (pictured below) will suck up the ants from the ground and collect them into a container. You can buy an aspirator from a science supplier, but it is just as easy to make one yourself. Below is one I use built from a cordless hand vacuum. The essential components are the collection chamber (plastic food storage container) and the mesh screening over the source of suction.
An aspirator will allow you to collect large numbers of army ants when you encounter them (see step 2). Collection of the evacuating ants also necessitates an aspirator, as they will be evacuating their nests rapidly (see step 4).
Step 4: Collect Your Ants!
Fifty army ants are enough to trigger a full evacuation. Once the army ants are poured in, Aphaenogaster workers will erupt out of their nests carrying their brood. If the queen is present she will also emerge, either on her own or being carried by a worker.
Aspirate the ants as they emerge and within 10-15 minutes you will have a complete colony, ready for observation and scientific discovery!