How do I test a circuit without an oscilloscope?
An Oscilloscope and multimeter is standard for electronics work, and not regarded as fancy laboratory equipment.
You can pick a second hand analogue scope up on eBay for a few dollars which will make your life a lot easier.  

To answer the question though, if your signals are audio then one easy way of testing is a small speaker/pair of headphones. If loading is a problem you can rig up a little fet amplifier.
Another option would be a bulb of some sort, a VU meter, etc. A multimeter on AC will give some indication of signal level (maybe not so accurate depending on quality of meter and signal frequencies.  

One cheap solution is to use a sound card and some free scope software. This will only be good for AC frequencies between ~20Hz to 20kHz (possibly a bit higher depending on sound card) but is certainly better than no scope at all.  

Visual Analyzer is a pretty good example of such a tool. If you hack together a probe of some sort (e.g. stereo headphone leads with shield and signal for right left channel = 2 probes)
The sound card input will probably only be good for up to around 2 or 3 volts, but a small opamp circuit can be used to divide/amplify and make it usable over a greater input range if necessary. I think there is an example circuit given to use with the software at the link above.