Transistor Logic Gates in One Breadboard
If you just want to make your own logic gates to see them work, then RTL (resistor-transistor logic) is probably the easiest for you to implement.

The basic gate output is a pullup resistor to the supply with a NPN transistor connected between it and ground.  When the transistor is on, the output is actively driven low.  When the transistor is off, the output is passively pulled high.

Use about 5 V for the power supply and 2 k&Omega; for the pullup resistors.  These logic gates will be very slow and power hungry by today's standards, but that is of no consequence when experimenting with a few of them on a breadboard.  You wouldn't want to build a CPU out of such gates because the power dissipation would be enormous, but you're not trying to do that so don't worry about it.  Note that each output that is low will draw 2.5 mA from the power supply, which is 12.5 mW.  There is no problem with this with a few gates on a breadboard.

Here is a simple inverter:



To make a NOR gate, put a diode feeding IN from each of the NOR inputs.  Note that this is a good example of NOR being simpler than OR with inverter on the output.

Get a 100 each of signal NPN transistors, 10 k&Omega; and 2 k&Omega; resistors, and small signal diodes, then start experimenting.