Shannaboo, here's the basics of it:
1. You need a computer with decent processing power. I'd say at LEAST 2.0 Ghz with a gig of RAM, and preferably a fast hard drive to save your audio on (the Western Digital Raptor drives are 10,000rpm and one of the best you'll find).
2. You need a soundcard with enough Inputs/Outputs to get all of your audio channels into the computer. Don't even bother with a gaming soundcard. You need a pro audio card (the M-Audio and Emu cards are good quality on a low budget... there are others too). If you're only going to be recording vocals and one instrument at a time, you really only need a soundcard with two channels. However, if you ever want to record a entire band or a drum kit (both of which require many channels to be recorded at once), you need multiple I/Os.
3. You need something to provide A/D and D/A conversion (analog-to-digital and digital-to-analog conversion). The A/D and D/A converters take the analog signals from your mic preamps and instrument DI boxes and convert them to digital format to be used in the computer. Many soundcards have analog inputs with onboard A/D/A conversion. Many mic preamps also have onboard A/D conversion providing you with digital outputs to go straight to your soundcard. Otherwise, you can get a stand-alone A/D/A converter. Either way, the signal must be converted to digital before it is recorded on your computer. NOTE: this is not an area you want to skimp on. The quality of A/D conversion has a HUGE impact on the quality of your recordings.
4. You need mic preamps and/or instrument DI boxes to convert your mic and instrument level signals to line level (which is what your A/D/A converters accept. This stage also has a HUGE impact on your final recording. A poor mic preamp can make your recording sound lifeless and muffled. Good mic preamps make your recording come alive, and can impart a certain "color" or quality to the track. Different preamps have different sonic characteristics. Some are very clean and pristine. Some are very warm. Some are dark. Some are aggressive, etc, etc. The songs you hear on Top 40 radio were likely recorded using mic preamps that cost between $750 and $2000 USD per channel. You can, however, get very good alternatives for much cheaper, especially since (I'm assuming) you're not looking to take on paying clients that expect to see certain name brands. I like to think you can get yourself 80% of the way to a high-end preamp on a low budget. Low-end mic preamps are an area of much debate and it seems everyone has their own opinions on which are the best bang for your buck.
5. You need microphones to record your vocals and instruments. Different microphones match up better with different instruments and voices. Vocals, for example, are usually recorded using a large diaphragm condensor mic. Guitar amps are often recorded using a dynamic mic (such as the Shure SM57). Like mic preamps, different mics will color your sound much differently. On a large budget, you'll likely end up using a different mic on every instrument. If you can only afford one or two mics to start out, I'd suggest getting a large diaphragm condensor for vocals and other acoustic instruments, and a Shure SM57 for everything else.
6. You said you'd like to add digital instruments, so you'd also need a midi interface and a midi controller. Many recording soundcards have an onboard midi interface with the proper I/Os. The most common midi controller is a form of keyboard. You can get a dedicated keyboard midi controller, but many stage keyboards also have midi capabilities. You'd then get software to provide the actual instrument sounds. There are MANY software programs out there with hordes of different instrument emulations.
7. You need a pair of studio reference monitor speakers for mixing and editing. You can mix using any speaker system (even an ordinary HiFi stereo), but to accurately hear any problems in your mix, you need a set of reference speakers with a flat response.
8. You need a software recorder and mixer. There are many many choices... ProTools, Logic, Cubase, Nuendo, Sonar, etc. They all do basically the same thing. Pick one with an interface that you are comfortable working on, and stick to it.
9. Effects... many many many possibilities here. It boils down to either hardware effects, or software emulations of hardware effects. Hardware effects units generally sound better, but are very expensive. Software effects, despite the vast improvements over the last few years, generally don't sound as good, but are MUCH more economically feasible. For example, the Fairchild 670 Compressor (a legendary hardware compressor) would cost you about $30,000 bucks if you were lucky enough to ever find one. With the UAD-1 dsp card, you can get their software plug-in version for $150, which will probably sound 95% as good.
These are the basics. Do some searches on this and other forums to see what other people are using, given your personal budget. There are endless combinations of usable gear. I'd suggest you do a lot of research before you make any purchases. Hope this helps.
Jay