How to make the the voltage higher after a logic gate
Here's a few ways of doing the same thing.

Circuit 1 - is a simple AND gate made from two switches in series. The LED is driven directly with R1 to limit the LED current.
Circuit 2 - replace the LED/resistor with a relay coil. Note the reverse diode across the coil which prevents the back emf spike from damaging the switches when you turn OFF. The switch contacts of the relay now control the LED.
Circuit 3 - Now we've added the 2 input AND gate. Note R1 and R4 (47k) are PULL DOWN resistors to ensure the inputs stay at 0V when the switches are open. This resistor value is not critical and could vary from 1k0 to 1M0. The output of the gate is capable of supplying enough current to light an LED. Again the 1k0 resistor is there to limit the maximum LED current.
Circuit 4 - If we need more current output we can add a transistor, in this case a BJT but we could use an N channel MOSFET (not shown) if we needed to switch a much higher current e.g. a motor or a solenoid. The transistor is just used to switch the LED.
Circuit 5 - The transistor is being used to switch the relay coil. Note we keep the reverse connected diode. In this case it protects the transistor. R3 value has been lowered to allow for more current through the transistor taken by a relay coil.