Implementations of circuits like this can vary somewhat, but I'll reference the diagram you gave:
C1, C2 and the inductor should determine the frequency of oscillation. Normally getting an accurate frequency is important, so these will probably be one of the more accurate types of capacitors (polystyrene, mica, polyester e.g.). Do you know how to analyse the circuit to find the frequency of oscillation?
You're correct in thinking that R1, R2 and R3 set the voltages (DC bias voltages) for the transistor. Have you ever seen a class A (one transistor) amplifier? Similar idea - you set voltages around the transistor so that it sits in a certain operating region.
The oscillations will be at the collector, and it's best to think of the voltage there as being the sum of a DC component (from the bias voltages) plus the oscillation.
Now the output voltage, should have the AC component of the signal (the oscillation) but not the DC - otherwise when you connect the output to another circuit, quite a lot of current will flow, and you'll waste a lot of power. How might the combination of C5 and R0 achieve that? remember that capacitors tend to have a low impedence to higher frequencies, and are an open circuit to DC.
C3 and C4 are decoupling capacitors (google it if you've never heard the term before). In short, they help keep the voltages steady in different parts of the circuit. Again, think about what capacitors do with higher frequencies (AC) as compared to DC to see how that works. Normally these will have a relatively large capacitance (maybe 10 to 100 microfarads) and be electrolytic, since the exact capacitance value is unimportant (and electrolytic caps are relatively cheap and inaccurate for their size).
Hope this helps.
