Spend some time in the processing threads on this site or other sites that cover basic post processing practices.
You have to set your recording levels in response to the peak levels at the show. Do not run them arbitrarily loud and clip things (you seem to have found that out but may not then be responding as you should). You can absolutely raise the levels of anything later but once something is digitally overrecorded it is almost impossible (or is impossible) to effectively repair. Leave some "head room" and then boost the output in editing. Peaks at 3 to 6 dB below 0 is a fair margin (but the exact amount is subjective).
In terms of raising levels in post you have many choices:
Amplitude (volume) just boosts the entire thing by whatever amount you specify. It is totally neutral so does not change anything except the perceived volume. All software should be transparent in doing that (though not sure if it all is).
Other techniques may change the sound (and certainly the dynamics) from slightly to significantly and the results will vary with the precision of the software.
Normalizing can be done in several ways. A simple explanation is here:
http://en.wikipedia.org/wiki/Audio_normalization In theory a neutral sort of basic normalization will automatically put as much output into your result as it can handle without changing the dynamics (giving you the highest potential output without altering any dynamics). Other sorts of "normalization" may change the dynamics.
The loudness war argument is valid, but generally refers to very heavy compression (or as the above wiki note terms it certain techniques of "loudness normalization") where clipping is introduced. It becomes the equivalent of running your recorder at a volume that is several dB's too high for the program (and then sounds like those lost to distortion).
There are also various forms of compression. Nearly all music releases employ varying degrees of compression and often several types together.
Hard limiting is a good option if you have a few spikes and the rest of the program is more or less in a fairly even range. A hard limit will boost the rest to a set level but boost the spikes less. You have to be able to run that in a way where you can preview the result to make sure you're not pushing the spikes over 0 db into clipping (or clipping them at whatever upper limit you set). It is sort of like the limiting function on the recorder but done consciously in post and hopefully done in a way to ensure nothing clips (whereas the limiters on most recorders are pretty harsh and do not prevent clipping - they just reduce the impact after something clips).
"Compression" generally refers to techniques that reduce the dynamic range in order to provide room to increase the average output level. Done properly it can enhance the listening experience.
There are different compression functions that will have various effects depending on the settings and the software. Use with care.
Not all software is equal. You often get what you pay for (or don't). Audacity is probably OK for volume adjustments, but at least some of the higher end processing algorithms are not up to the standard of commercial programs.
The musicians were right that you want to back your levels under the point of clipping and then when processing it afterward add levels in a way that also remains under the point of clipping.