Getting the sounds you want out of an analog synth take as much practice as playing the keyboard unless you have a real understanding of it. It may be hard to imagine an original synthesizer sound when professionals with omnipotent studios at their disposal can't stop copying 80s club music and using normal filter effects but keep reading and you may feel a glimmer of hope!

HARMONICS
Mimicking acoustic instruments is not the goal of analog synthesis but understanding how they work is a real key to generating the sounds you want. A harmonic, which is also called an overtone, is supposedly an integer multiple of the main frequency of a sound. There may be harmonics going on into infinity in frequency but most won't hear them above 20 thousand hertz, nor pure tones below 20 hertz. Some harmonics can be a little sharp or flat and still be considered harmonic rather than inharmonic. Partials include these harmonics in most texts and also any other frequency that makes up a tone. With a waveform or a musical instrument there is still a fundamental, a frequency which is dominant and usually the lowest. When you use a waveform other than a sine wave it is considered to have harmonics even though it appears to be a single frequency, which makes harmonics and partials hard to understand. It is interesting to note that by adding sine waves together which can be generated and mixed normally acoustic sounding signals have resulted. Even electric organs use partials, for example C at 126 HZ and G at 190 HZ at once will get a difference tone between them of 190-126=64 cycles per second which is C in a different octave. This is why fifths sound the way they do. Getting a sawtooth wave through additive synthesis only requires about 14 sine wave harmonics added together. Once added together they should no longer be visible on an oscilloscope if they are in their proper position or phase.
Different sounding tones often have the same harmonics at different volumes. You can alter them with a filter or equalizers. Harmonics can be measured in decibels and HZ as is sometimes displayed on a digital equalizer, and remember that equalizers and filters can effect any frequency, even the fundamental. You can hear just harmonics if you eliminate the fundamental. In strictly electronic terms altering harmonics like this does not change any frequencies but warps the waveform with the same result, so harmonics may be considered abstract to some degree but they are still quite real. Using the filters resonance will show you on an oscilloscope that a "ringing" can occur purely through emphasis on a harmonic area. See the filter and phaser page for more.
Imagine a bugle that has no stops and no keys, how are different notes produced? Understanding harmonics gives you the answer to some degree. If you know how to get a harmonic by touching the node of a stringed instrument like a guitar or bass you can see that different harmonics are like different notes.
Quoting Bob Moog "There is no such thing as a piano spectrum because the amplitudes of various piano tone harmonics are constantly changing with respect to one another in complex ways." He goes on to describe how the piano's harmonics gradually go sharp as the note goes on while the fundamental remains the same. Acoustician Harvey Fletcher made an additive synthesis piano sound with a hundred oscillators that was similar to the acoustic sound.
Combining 2 waveforms can show you a bit about harmonics. An electronic organ's bars are pretty much adjusting the volume of those harmonics and a ring modulator creates unusual partials to make a harsh or pitchless sound.

Combining 2 Signals
Here are some ways to make 2 signals into one with aspects of both. Mixing 2 sounds together is the obvious way, you can mix several octaves of sine waves to simulate the harmonics of other waveforms or mix tones that or not quite at the same note to get a chorus like beating effect. Mixing two sounds and then applying extremely heavy distortion, or even some type of sharp clipping, can produce some rarely heard effects often similar to ring modulation. Of course this effect occurs without distortion but it is very smooth, and of course you can also hear it with an acoustic instrument. Distorting a sound can make it more like a waveform. Ring modulation is an amplitude modulation effect that modulates positive and negative resting at 0 output when one of it's inputs is quiet. The vocorder effect makes one signal mimic the treble and bass character of the other. Oscillator sync can only function with 2 oscillators. If you chose instead to use a distorted guitar signal, distorted being more like a square wave, as the soft sync master you would get a weird effect when you played a chord or solo. You would also get some funny sounds if you used square wave chords as the sync master. The master is the sound that determines the overall pitch. There are also various forms of amplitude modulation, frequency modulation and filter modulation as well as pulse width modulation for square waves that combine aspects of the modulator with the main signal. With analog synthesizers you can use any sound source as a modulator provided it is loud enough. If you spoke into a microphone and fed that signal only to the fm input of a VCO you could probably get that VCO to make words.
Just mixing different oscillators with different settings can get the most unusual sounds and they often sound like it is just one note. When the first digital synths used waveforms to imitate a piano they tried tricks like having one oscillator make a click like a key and hammer mechanism and two or preferably 3 oscillators making string sounds to mimic each string in one piano note. You don't have to mimic something to use this method however.
Drones make a really unusual style of music. It imitates instruments like bagpipes, harmoniums, and snake charmer's pipes but it doesn't have to be acoustic. A drone is a constant note played underneath a melody. A drone can be continuous or it can go quiet when the melody goes quiet, and the best method is combining the two. You usually do this with a sequencer. Modulating a drone's pulse width or cutoff with an lfo is good but try some other methods like modulating it slightly with the melody notes or the cutoff with the melody's keyboard tracking.

Keyboard tracking makes for more variety.
Most synthesizers have a resonant filter. Even if your synthesizer is a new digital powerhouse of samples rather than an analog one it still should have a conventional filter like those that made the first synthesizers so versatile. What you may not realize is that most of the time you are using it and it may be adding a certain frequency to the sound if the resonance and cutoff frequency are set right. Sometimes there is a keyboard tracking adjustment that can shape the way the filter frequency behaves depending on the note the synthesizer plays. The note and the filter cutoff frequency can be kept the same at all times if you like but always try all the different was of setting it.
The cutoff does make a more even sound with the keyboard tracking but tone is all about getting a more varied sound out of a normal set of notes, so setting the tracking in an unusual way or turning it off can help mimic musical instruments or enhance a bizarre waveform. It's also really good to use other things besides just cutoff. Keyboard tracking pulse width or ring modulator pitch really makes a difference. If you play a normal note ring modulated with a note less than a semitone off it makes a unique sound, so set the modulator pitch's keyboard tracking up so it's just a little off.

Bad Sounds Good?!
A sideband is a partial, a sine like frequency that is a component of a sound, which is not in tune with the normal harmonic spectrum. Just mixing two notes together can produce a sideband electrically or acoustically. In some cases you can tell what these frequencies will be mathematically.
Amplitude modulation creates sum and difference frequencies. Adding the number of HZ together and subtracting the lesser from the greater should give you a good approximation. Filter modulation, especially with resonance, causes more complicated sideboards and frequency modulation as well as pulse width seem to cause the most. When these sidebands are made louder by heavier modulation the fundamental of the original waveform is overpowered. If the modulating signal is out of the audible spectrum you may still hear a few sidebands.
When you use vibrato it seems like it just changes the pitch but it also creates these sidebands thereby effecting the musical nature of the sound more than you may realize. If you look at a graph of it you will see it actually warps the waveform more the closer the vibrato speed is in relation to the pitch of that waveform. If you take a waveform and feed it into it's own frequency modulation input on an analog modular synth you simply get different harmonic content and a different waveform.
The most nonsensical sounds can become sweet melody if used right. A bell can easily have more powerful inharmonic sidebands than it's fundamental and harmonics. This can make it more effective at alerting people of something because it's easier to hear above other noises. This is not the kind of bell that you normally use for music because of it's inharmonious sound although this is a matter of opinion. If you like modulated or toneless sounds but want more emotion and melody to them try finding ways of sequencing and looping them, use the same types of patterns you use for melody or bass lines. Try an experiment with a ring modulator. First make it sound really weird and then play a repeating bass line with it. The tune has no recognizable notes but you may notice a melody too. Alternating from noisy to melodious makes a good balance and so does combining them together. Try playing a normal repeating melody and have a heavily modulated bass line. Then make the melody heavily modulated and the bass line normal and musical. Noise solos and modulated arpeggios are fun even if they don't fit into melody things too well.

Getting strange effects
If you have a multi effects processor foot pedal or rack module you can have a lot of fun with it with a synthesizer even if it is made for guitar. Often the drawback is that the frequencies are tailored in a way that eliminates some bass or boosts middle harmonics etc. but just try it and you shouldn't cause any damage. Digital effects may not be your thing but often the analog distortion and compression can make some interesting variations. Never underestimate distortion, it got all that fame for guitars but it works on any signal.

Analog drumkit sounds
Back when you were combining 2 signals you may have thought about getting drum sounds. This is best done with a specialized drum module but don't let that stop you. A normal sounding drum can be made by using an envelope to control the pitch of a triangle wave an bending it down 2 or more octaves. It will loose it's quality of pitch and become percussive, and you can change the filter and pitch to get toms or a bass drum. If you bend it up from a low note it will be an open tom or hand drum. When you have a sound you like you can enhance it with another oscillator that has a shorter duration to make the drumstick on the shell sound. This tone should also bend down but even quicker than the first. The snare can be done with this method by adding a third enveloped white noise sound. If you don't like the result try frequency or ring modulation on the triangle wave that does the bend. Modulating it with noise will make a splat sound but if done right it will sound like a weird snare. Noise used as a modulation source can make a good high hat tone sometimes.