What you need
-A bunch of 5k and 10k resistors, and transistors
You can use any NPN type transistor (for example 2N3904, BC547, BC548, BC549 etc.)
These are basic circuits, so they can be improved (with diodes for reverse voltage protection, for example), or can be simplified (XOR, XNOR gates), but for start we will stick with these basic circuits.
These circuits are extremely easy to understand. I think if you know a little bit about NPN transistors, you will be fine. The use of transistors for the construction of logic gates depends upon their utility as fast switches. When the base-emitter diode is turned on enough to be driven into saturation, the collector voltage with respect to the emitter may be near zero and can be used to construct gates for the TTL logic family.
I am going to show the logic gate's symbols, truth tables and schematics. I will also show this circuits on the breadboard. (As i mentioned before these are easy circuits, so on a breadboard I wont show them step-by-step) There is a small text and a gif for every logic gate.
I attached three images for beginners. The third image is the most important.
Step 1: NOT Gate
This is the easiest one.
For the NOT logic, there is only one transistor and the output is driven to the ground (LOW or 0) if it is conducting.
Step 3: AND Gate
For the AND logic, the transistors are in series and both transistors must be in the conducting state to drive the output high.
Step 5: NAND Gate
For the NAND logic, the transistors are in series, but the output is above them. The output is high unless both A and B inputs are high, in which case the output is taken down close to ground potential.
In the simulator I accidentally used a 200Ohm resistor, but in reality you should use around 1K or something like that.
Step 7: OR Gate
For the OR logic, the transistors are in parallel and the output is driven high if either of the transistors is conducting.
Step 9: NOR Gate
For the NOR logic, the transistors are in parallel with the output above them so that if either or both of the inputs are high, the output is driven low.
In the simulator I used a 200 Ohms resistor, but you should use one around 1K.
Step 11: XOR Gate
As you can see I constructed this one with other gates. In this circuit I used a NAND, OR and an AND gate.
Step 12: XNOR Gate
As you can see I constructed this one with other gates. I used two inverter, two AND and an OR gate in this circuit.
Step 13: Build Something! - SR Flip-flop
When using static gates as building blocks, the most fundamental latch is the simple SR latch, where S and R stand for set and reset. It can be constructed from a pair of cross-coupled NOR logic gates. The stored bit is present on the output marked Q.
The output usually marked as Q, but sometimes it is marked as X, Y or F.
While the S and R inputs are both low, feedback maintains the Q and Q' outputs in a constant state, with Q' the complement of Q. If S (Set) is pulsed high (=when you push the button) while R (Reset) is held low, then the Q output is forced high, and stays high when S returns to low; similarly, if R is pulsed high while S is held low, then the Q output is forced low, and stays low when R returns to low.
Now we are ready, you made an SR flip-flop or SR latch.
When you press the button one, the Q LED goes out, and the Q' LED lights up. Then if you press the button two, the Q LED light up, and the Q' goes out. Technically you can store data with it, this is why we call it "latch".
You shouldt press the two buttons at once, its a forbidden state, it can make the flip-flop unstable.