Low Frequency, High Voltage Oscillator

Yes, that was Matt who told you about the filtering to get a sine from a square wave; unfortunately he forgot to tell you how to do this. :-)
Well, it's not easy. A 1kHz square wave will have a 3kHz harmonic with 1/3 amplitude, a 5kHz with 1/5 amplitude, etc. That's a lot of energy in the harmonics which you want to get rid of. You need a sharp low-pass filter to be sure that the 3kHz is attenuated enough. That's possible, but not so easy to make it variable; for the 2kHz sine you want a higher cut-off frequency.

The plot shows that the signal still looks more like a square wave than a sine even with all except the third of the harmonics removed.
All this suggests that this may be an easy approach to generate different crystal-stable frequencies from a single 12MHz oscillator, but not sines. For sines there are analog solutions like the Wien bridge, but these aren't so frequency stable.

Like I said in my other answer DDS really is the way to go; you get the best result for the money. If you don't want to use a microcontroller you can use a lookup ROM programmed with a sine waveform, let a counter run over all addresses, and feed the output to a DAC. That requires a little bit of logic, but no uC. The clock you use for the counter will be a multiple of the sine, so feeding a different clock frequency will give you a different sine frequency.
All this said, this may cost as much as a microcontroller; there's no reason to dismiss a microcontroller because it would be too expensive. uCs are dirt cheap these days and often a more economical solution that analog alternatives. You won't find anything simpler than Jesper's generator (was linked here http://www.myplace.nu/avr/minidds/index.htm, link now broken)

If you want to generate the 5 signals simultaneously and mix them the DDS solution is even more cost-effective: you'll need only one (1), vs 5 oscillators in an analog solution.

edit
If you only need 5 fixed frequencies you might want to switch them on and off selectively, so that you can mix them. Easy with the DSS uC: instead of keeping one phase accumulator (expensive word for "counter"), you just keep 5 of them, and add the sine values before sending them to the external DAC. You could use 5 switches to turn them on and off. You don't need the MAX232 then. Like I said: cheap.

Yes, the 16F84 does need an external oscillator. It is a very old PIC.

However almost all of the newer PICs have an Internal RC Oscillator that can be selected, which will be mentioned in the datasheet.

I would really consider getting hold of a newer PIC, something like a 16F690, or 16F1824/16F1828. These are far more current, and can do anything the 16F84 can do and much more.

If you want to use your 16F84 though, either use an external clock (e.g. from 555 timer or oscillator based on e.g. an inverting gate with RC or crystal) or crystal as specified in the datasheet, or if you don't have an external clock or crystal use the RC option.