Since your question is very general, I can only guess as to what guidance you actually need. Here's a start.
Due to my inexperience with op amps and audio signal design, what would be a good way of amplifying the signal, and what would be a good way of filtering the signal?
Amplification and filtering often go by the single term signal "conditioning".
Given that the human voice has a range of about 300Hz to 3.4kHz (according to Wikipedia), it would seem that a bandpass filter would be sufficient to reject frequencies outside this range. An active bandpass filter, one with active electronics (usually op-amps), allows you to amplify your signal as well (since op-amps are just that, amplifiers). This could take some of the weight off a separate amplifier circuit, which you may need to boost your signal to usable levels.
Here's a good link to active bandpass filter topology and analysis. This link makes good arguments for both cascaded filter design (first section) and multiple feedback design (second section).
Cascaded design uses three stages which combine to create an active bandpass filter: a highpass filter, an amplifier, and a lowpass filter. This approach gives you a very flat passband, which makes it good for wide ranges like our 300Hz to 3.4kHz. If you implemented this method, I would perform the lowpass filtering first, because the most prominent noise in your system will likely have high frequency content, and you don't want to amplify that before filtering it.
The multiple feedback design also has its pros. For example the issue with doing lowpass filtering first (described above) is negligible. However, it is best used for narrow passbands, which 300Hz to 3.4kHz is not. I'd recommend the cascaded approach above.