The LiveAmp is indeed a good choice for ERP research but needs to be combined also with the Sensor and Trigger Extension, in order to be able to accept complex (8-bit triggers).
These triggers can be sent either via trigger cable or via Wireless Trigger.
The box on the right hand side is the TriggerBox, which is a handy piece of equipment when it comes to receiving triggers from different sources and forwarding them to the EEG amplifiers. In the most common case, the TriggerBox can receive triggers from the USB port of a stimulation PC (running e.g., E-Prime) and forward it to the STE of the LiveAmp. It can also receive triggers from a LPT port or from BNC inputs.
In terms of electrode technology, we usually recommend to use it with actiCAP slim, since the active electrode technology works great for mobile studies since it reduces the chance for environmental noise. Moreover, setting up the cap is quick and fun since the electrodes have LEDs that give coloured optical impedance level feedback (red, yellow, green). Setting up a 32 channels cap take around 4-6 minutes when bringing the impedance down to 25kOhm, which is our recommended level for great data quality (yes it is higher than the classic 5-10kOhm recommended for passive electrodes).
In alternative, if your participants are not going to move too much (e.g., if they walk normally instead of running), you can also use the R-Net, which, instead of gel, uses sponges wet in a salt-water solution to create a contact between the electrode and the scalp. Set-up is gel free and quite fast (1-2 minutes for 32 channels) but impedances will be higher (~100kOhm) and therefore the bandwidth will be restricted to 0-100Hz since the signal would get noisier at higher frequencies. Nevertheless, such bandwidth is good enough for 90% of the ERP applications.