I can't really make heads or tails of your schematic but if you want to test the 74LS86 you can do the following:
Connect pin 7 to ground
Connect pin 14 to +5V
You are going to test 1 gate at a time.
For the first gate:
Connect input 1A to +5V through a 10 kΩ resistor.
Connect input 1B to +5V through another 10 kΩ resistor.
Connect output 1Y to ground through a third 10 kΩ resistor.
Connect all the remaining inputs to ground (2A,2B,3A,3B,4A and 4B).
Leave the remaining outputs unconnected.
Set your DMM to measure DC voltage and connect the positive lead to the 1Y output (pin 3). Connect the negative lead to ground.
At this point you can connect the voltage souce.
Now if everything is hooked up correctly at this point, both inputs (1A and 1B) will be high so the output (1Y) should be low (≈ 0V). Check your DMM to make sure.
A=H, B=H, Y=L
Next, connect 1A to ground and then check your DMM. It should be ≈ 5V.
A=L, B=H, Y=H
Disconnect 1A from gound and connect 1B to ground. Check your DMM, it should be ≈ 5V.
A=H, B=L, Y=H
Next, leaving 1B connected to ground, reconnect 1A to ground and check your DMM. It should be ≈ 0V.
A=L, B=L, Y=L
That's it. You just tested every possible state of the 2-input XOR gate.
Repeat the above steps for the other 3 gates making sure to ground the unused inputs.
I hope this makes sense.