The circuit below is a standard oscillator of the Colpitts variety. Similar circuits have been used in many ham radio homebrew transmitters. This particular circuit should function well at frequencies from 1500 kHz to 8000 kHz. For use on lower frequencies, the values of C1 and C2 might need to be increased.
C1: 100 pF ceramic disc or silver mica
C2: 680 pF ceramic disc or silver mica
C3: .01 uF ceramic disc
C4: .001 uF ceramic disc
R1: 220 K
R2: 1 K
beginner's assembly instructions:
If you've never built a circuit from a schematic before, this might be a good one to start with. The diagram below shows how you can arrange the parts on a prototyping board such as Radio Shack catalog number 276-175. (A prototyping board, also called a solderless breadboard, contains groups of holes that are electrically connected. Each hole has a little spring/clamp thingy in it that grabs ahold of the component leads. This is a great way to experiment with circuit designs and learn about the building process.)
Attach a 10-inch (25 cm) piece of bare wire to the output, then sit a radio next to the circuit and tune to the crystal frequency. Apply power. If everything is connected correctly and all the components are in working order, you will hear the carrier (or the silence caused by it) on your receiver. Now all you need to do is add a buffer amp, a modulation stage, a final RF amp, a harmonic suppression and output matching section, and you've got an AM transmitter. :-)
Below is another version of the circuit with a couple of enhancements. The variable capacitor between the crystal and ground allows you to adjust the frequency slightly. (More capacitance equals lower frequency.) Q2 serves as a buffer amplifier which stabilizes the circuit and boosts the output power. This circuit was developed independently by another MWA member and uses very different values for R1, C1 and C2 compared to the first circuit on this page; don't let those differences scare you.
C1, C4, C5: .001 uF
C2: .0033 uF
C3: 20 to 50 pF variable
Q1, Q2: 2N3904
R1: 22 K
R2, R6: 1 K
R3: 18 K
R4: 270 K
R5: 470 K
Another common variation of the Colpitts circuit involves adding a resistor parallel to the crystal, as shown below. What's the advantage? I have no idea. The circuit shown has been tested and works fine at frequencies from 1.5 to 20 MHz.
To round out the collection, here's a Pierce oscillator using an FET. The version on the left is from an electronics textbook. The version on the right is from the "Grenade" shortwave pirate radio transmitter designed by "Radio Animal."
Some people might prefer an oscillator that uses a crystal at 4 times the operating frequency and then divides by four to produce the carrier wave. Advantages are 1) your signal will have a super stable frequency with immeasurably low drift, and 2) you can order a custom-made crystal without it being obvious to the manufacturer that it will be used for a broadcast-band application. The disadvantages are 1) the circuit is more complex, and 2) the output of the oscillator will be a square-wave rich in harmonics. If you would be interested in this approach, you can consider building a circuit inspired by the design of the oscillator section of the Wild Planet toy transmitter.