Russell gave a great overview of various logic families, and you should explore the many references in his answer.
In terms of actually building some circuits and experimenting with them, I suggest using DTL (diode transistor logic), which along with RTL (resistor transistor logic), were two of the early logic families used successfully used in several large projects in the 1960's, like the Apollo guidance computer (RTL), IBM 360 (DTL), and Minuteman II guidance computer (DTL).
DTL has some performance improvements over RTL (such as greater fan-in). On the other hand, it is simpler than TTL (transistor transistor logic) circuits for the same function. (Logic designers quickly switched over to using TTL for integrated circuits because its easy to implement transistors on an IC.) All of these (RTL, DTL, TTL) used BJT's (Bipolar Junction Transistors). They were later replaced by CMOS using FET's (Field Effect Transistors), which use much less power.
The two building blocks of logic circuits are the NAND gate and the NOR gate. All other logic functions can be made out of either of these two. (For example, the Apollo guidance computer used 5600 3-input NOR gates and nothing else to implement all of its logic.)
The truth tables for the NAND and NOR gates aare:
NAND NOR
A B Out A B Out
------------ -----------
0 0 1 0 0 1
0 1 1 0 1 0
1 0 1 1 0 0
1 1 0 1 1 0
Here is a simplified NAND gate:

If either or both inputs are a logic 0, i.e. ground, one or both diodes will conduct, so the base of the transistor will be at the V\$_{f}\$ of the diodes, or about 0.7v. This will not be enough to turn on the transistor (well, very close to V\$_{BE}\$, see below) so the output of the transistor will be near +5, or a logic 1, as shown in the truth table above.
If both A and B are logic 1 (+5), then neither diode conducts, and 5v is applied to the base of the transistor, turning it on. This then puts a ground (actually V\$_{CE}\$, or about 0.2v) on the output, representing a logic 0.
In practice, one or more diodes are placed between the input diodes and the base of the transistor, to set a higher threshold to insure the transistor doesn't turn on when either input is low. In addition, a resistor can be placed between the base and ground, so the transistor turns off faster when the base current is removed.
Here are some DTL circuits for implementing NAND, NOR, AND, OR, and NOT:

You can use the ubiquitous IN4148 for all the diodes, and any general purpose NPN transistor such as the 2N3904.
Note that the AND is just a NAND with an inverter tacked on, and the same for the NOR and OR.
These circuits are adapted from this document, which includes an exhaustive circuit analysis of the NAND gate circuit.
All of the circuits are designed so the outputs of one gate feed into the inputs of another.