Re: microphones for a saxophone
With regard to a PC recording setup, there are a variety of ways you can go, depending on what your budget and objectives are and how controlled your recording environment is. For basic playback of your practice with reasonable fidelity, the simplest and most economical setup is a USB mic that plugs directly into your PC. If you're going to be recording in a less than perfect acoustic environment, a gain control or pad switch is a good feature. A headphone jack is good if you're going to record yourself over backing tracks. There are some good ones with either or both of those features available for between $100-$200. Here's a place to get started in selecting the USB mic for you.
The most full-featured mic in that range is the Blue Yeti - multiple pick-up patterns, gain control, zero latency headphone jack. Quite reasonably priced.
The Audio Technica AT2020 USB Plus doesn't have the gain control or the multiple pickup patterns, but it has a larger condenser capsule that yields a little warmer recording. Costs more than the Blue Yeti, but still a good deal.
If you're all about recording quality and don't mind doing without a headphone jack, the AKG Perception 120 USB is a good option. It has a -20dB pad switch and 24 bit D/A converter, while the above two have 16 bit converters. Price is comparable to the Blue Yeti.
If you don't mind breaking the $200 barrier you can get all of the features of the Blue Yeti, plus 24 bit D/A fidelity and an XLR output (so you can use it with mixers and such) with the Blue Yeti Pro. Definitely one to consider if you think you might use your mic for more sophisticated recording in the future.