Electronic – What makes a mic preamp good or bad

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.

Using a pot as a voltage divider as you do is not a good volume/gain control. The problem is that it makes your output impedance dependent on the gain. This has the effect of making your frequency response vary with gain as that output impedance interacts with the capacitance/impedance of your cable and whatever you are feeding this output into. You want the output impedance to be as low as possible. Do this by deleting R5 and R6.

To do the gain control without impacting the output impedance use a pot to replace either R3 or R4 (or both).

The nice thing about this type of gain control is that it actually controls the gain. With your configuration if the opamp was clipping and you turned the gain down you would still be clipping the opamp. But with the modified gain control, if you turn down the gain the opamp will stop clipping.

Do not remove C2, since it blocks the DC signals to your sound card. You might even consider increasing the size of C2 to as high as 68 uF. The larger C2 is, the better your low frequency response will be. 10 uF is the lowest I would go, but 22, 33, 47, or 68 uF are better choices.