The absolute essentials for home studio recording are:
A good mic. You will never get a natural sound out of the piezo element of your guitar alone; even the best ones sound plastic-y. You don't need a Neumann, but that $15 Radio Shack desktop mic ain't gonna cut it either. I would start looking around the $100/mic range for a dynamic mic, with the SM57 being a mainstay for high-gain sources like speaker cabinets, horns and mid-pitch drums, and not bad for vocals (the SM58 is more vocals-oriented, with a "mid-hump" in the 2kHz-6kHz range to emphasize diction). It'll work for guitar, but I recommend a condenser especially in a studio environment.
For condensers, which are best for moderate-volume instruments and high-detail studio vocals work, they run all over. My current faves for acoustic guitar on a budget are the Rode M5 matched pair of small-diaphragm condensers; they have the fast response you need to capture the fine detail of an acoustic guitar's sound, and they're half the price of Rode's flagship NT5 pair (the go-to for many, unless you have the money for a $1600 Neumann KM184 matched pair), while only being noticeably inferior at the extremes of their performance envelope (a little more self-noise, a little less sensitivity and max SPL).
For a do-it-all microphone, check out the Rode NT2a; it's a multi-pattern large-diaphragm condenser, good for vocals, guitar, just about any single source you want to point it at. LDCs as a rule don't handle "off-axis" sources as well as SDCs do, and they don't respond as quickly to transients, so they're not as faithful on things like drums and "percussive strings" like guitar and piano, but they do work for these sources, and they're excellent for vocals, bowed strings, and ambient setups capturing more room tone. You can buy these in a "Studio Essentials" kit including shockmount and breathscreen for about $400.
A good "interface" to your computer or digital recorder. Computer sound cards, no matter how good, usually can't take an XLR input. The mic input on most is designed for a computer microphone, which has an unbalanced signal with power sent over the "ring" terminal of the plug to the mic's electret. You will at least need a mic preamp designed to accept an XLR input, that will boost the gain to line-in levels that the sound card can then convert to digital. Better yet, invest further in a USB, Firewire or Thunderbolt audio interface that has the ADC built in and sends the digital signal directly to the OS.
On my desk at home right now is the Focusrite Scarlett 2i4, which I arrived at after much research on features and transparency versus cost. You can do much better, but you can also do much worse, and for the price it's excellent in features and sound quality; it's basically an outboard sound card, with hookups for two mics or line sources, a MIDI source like a keyboard or drumpad, and up to four output channels, two of which can be volume-controlled directly from the unit, as can a pair of headphones and the mix they receive between playback from the computer and the direct inputs.
For more inputs at one time, you start getting into the range of USB-capable mixers, like the Allen & Heath ZED series, which will do a mixdown of whatever you have connected, creating a stereo input to the computer. This can be good for quick recordings of groups, including live music situations, but they're expensive and not as good on the digital side as the dedicated interfaces.
A set of studio reference monitors. Consumer-grade headphones or speakers, even expensive brands like Beats, won't work very well; they typically have an overall "colored" response curve emphasizing either bass or high mids, and your cheaper ear buds and desktop speakers have reduced bass response. "Studio reference" means a flat response curve all the way from at least 20Hz to 20kHz, with some of your better ones able to accurately reproduce waveforms from 7Hz up to 30KHz (beyond the range of human hearing).
For the absolute bare essentials while recording, a pair of monitoring headphones will work, and you can get excellent ones in the $100-$200 range, with a few diamonds in the rough down to about $50. You will want these while tracking, especially if you want a mobile rig. However, when you're back in the "booth" doing your mixdown and setting up a stereo image for your final track, you will want a pair of free-standing reference monitors. These aren't cheap; you can expect to spend $150 a pair at the absolute low end (M-Audio AV40s), with most of the "good for the money" examples like Yamaha's HS series and the Rokits going for between $200 and $400 per speaker.
Software. Most people start with a freeware DAW, and there are many, most of them being lite versions of a payware DAW (such as SONAR LE, MuLab Free, etc), though there are a few open-source/freeware DAWs like Audacity and Traverso for Windows, GarageBand for Apple, and LMMS, Ardour and Qtractor for Linux.
I've used Audacity, and it's completely free and so good for a first step, but its lack of ASIO support out of the box (you have to download Visual Studio, the Audacity source code and the ASIO SDK, and build your own copy with ASIO support included; they can't distribute a compiled executable under the GPL with ASIO built-in) and near-lack of MIDI support (no MIDI recording or playback; all you can do is import and use the software synth to convert to an audio track) make it unsuitable for anything beyond the very basics of recording, editing, mixdown. If you want to add drums, or piano, or harp, or similar instruments to your recording, you have to record said drums or piano as a direct performance from whatever you're using to produce those sounds, and that makes it that much harder to edit the performance to tweak timing, edit note velocities, and other stuff you get more or less for free with a better DAW.
Professional paid apps topping the popularity charts include SONAR, Pro Tools, Reaper, Logic Pro, Ableton Live and FL Studio. Reaper's the cheapest for home amateurs and weekend warriors at just $60 (that's discounted pricing applying to personal use, nonprofit companies and small businesses earning less than $20k/yr), and has a hardcore following in that demographic. Pro Tools is at the other end of the spectrum, originally aimed at high-end production companies using the software alongside Avid's hardware offerings, but they also offer the standalone software for both Mac and PC and it's probably the first one you think of when you think DAW. Ableton Live is the grown-up version of the "Live Lite" software bundled with many audio interfaces. SONAR is dominant among working professionals running Windows, while Pro Tools is battling it out with Apple's own Logic Pro for Mac users.
A good setup. You can spend thousands of dollars, even tens of thousands, on quality gear, and get crappy results because the gear isn't set up right. The biggest things to get right are:
- To "use the symmetry of the room". Translated - choose a wall of the room that's relatively narrow, with the adjacent walls both perpendicular to it, with as few interruptions like doors and windows as possible on the adjacent walls, and set your desk up at the exact center of that wall.
- To set up the speakers forming a level equilateral triangle with your head. The speakers should be at head height, pointed directly at your head, and the distance between them should be equal to the distance you are from either one. This gives you the truest stereo image.
- To baffle the "angle of first reflection". To make sure you're hearing the left channel only in your left ear, and the right channel in your right ear, you have to make sure the sound waves from one speaker can't bounce off the opposite wall and hit your other ear. This is done by installing acoustical baffles at the point on the wall where those sound waves would bounce on their way to your opposite ear. The quick and dirty way to do it is to have someone hold a mirror at head height and slowly move it backward while you sit at the desk, and the point at which you first see the cone of the opposite speaker is the center of the point at which you should install baffling.
- To install bass traps as necessary. A lively room can accentuate bass frequencies due to "boundary coupling" of the speakers to the wall behind, and in other ways. Bass traps are foam blocks that fit in the corners of a room and soak up bass frequencies, so the bass you hear is what's naturally in your track.
- To isolate microphones from other sources of sound as much as possible. Replace that big, overclocked, 8-cooling-fans-droning gaming PC with a quieter desktop tower, or a laptop, to reduce computer noise. Choose a space in your home that's as removed from the rest of the house as you can get, and invest in a foam-core interior door and possibly even some weatherstripping to reduce air transmission through the door into the studio where you have live mikes. Get yourself an acoustical surround to use with individual mikes or coincident pairs, to reduce the amount of rear reflection the mic picks up.