Serial1 and Serial2 have hardware FIFO queues (first in, first out) that help reduce the requirements that the Teensy be ready to accept an interrupt when a character is available. If you expect to receive a lot of input, you want to use those two serial ports. The other 4 serial ports do not have the FIFOs, and so won't be able to be receive input as fast.
In addition if the hardware you are connecting to the Teensy supports RTS (request to send) and CTS (clear to send), you want to use that. Trasnmit enable, RTS. and CTS are hardware flow control bits that controls the flow of data if the Teensy cannot process it fast enough. Note, Serial4 does not support using CTS.
For each serial device the pins for receive, transmit, and CTS are fixed. RTS and transmit enable can be any digital pin. Serial1 can have alternate pins for receive, transmit, and CTS (but these pins are fixed also). You can get more information from this page:
So for Serial2, pin 9 is receive, pin 10 is transmit, and pin 14 is CTS. Note, that the Teensy only transmits voltage at 3.3v. This means if the device is expecting 5v, it may or may not see the 3.3v signal. The Teensy 3.6 is
NOT 5v tolerant, so you must make sure that voltages coming to the Teensy 3.6 are 3.3v, via level shifting if the voltages are not 3.3v.
Depending on how the device is documented, you typically would connect the RX pin of the Teensy to the TX pin of the device, and TX pin of the Teensy to the RX pin of the device. But some devices try to be 'helpful', and their RX/TX pins are swapped, so you would connect RX->RX and TX->TX.
Even if your device is powered separately, you will need to make sure the Teensy and the device share a common ground.