First of all I think it would be sensible to purchase (or get from library) a book on recording. I don't know what I would recommend but there are a fair few around. Amazon has user reviews and with any luck someone else will recommend something.
Reading up on internet is also worth doing.
If you are going down route of computers, free software is the way to go. I see that stuff and notice I have paid money for something which does exactly the same as that. Effects are expandable anyhow.
Here is an extensive list of free software
However, you will still end up spending a fair bit of money on hardware and even with budget sollutions, you are going to spend an amount of money that could get you a notable about of time in a (decent) studio.
You will need the following things for a computer based setup
software (free in this case)
microphones
microphone pre amps
A/D converter - recording interface
speaker monitors to hear what you record and/or headphones
For microphones, 1000s posibilites although it depends on your budget and what you want to record. Maybe best to start with something like Shure SM57 and expand collection later. If you don't get into recording, it wont become redundant or can be sold.
Not all voices are terribly suited to be recorded with condensors but some are. Some budget condensor microphones include AKG C1000, Studio Projects B1, Samson CO2
Converter/ recording interface - There are 100s of devices which allow you to record a few channels at once and sometimes include a microphone pre amp.
If you only want to record one or two instruments at a time (which it sounds like you do). There are various recording interfaces made from around the £50-200 mark which may be suitable. Companies to look for such things from include, M-Audio, Presonus and EMU. The cheaper devices usually don't sound great though.
If you wanted to record an entire band, including drums, something like the Presonus Firepod (renamed to FP10) would be worth concidering. It allows up to 10 input channels through firewire and contains 8 microphone pre amps. It typically goes for around £280.
pre amp - often there is one on your recorordng interface but you will need a microphone pre amp per microphone you record or else the microphone is recorded at an unsuitable level and will never sound 'right'. The ART MP tube is a popular choice for a budget microphone pre amp.
speakers - most people will tell you to get studio monitors. A set of good head phones might be sufficent but most people prefer to mix with speakers. Typical headphones for CD players generally wont cut it for this purpose. You are also better recording voice with headphones as no sound will leak into microphone.
Many recording interfaces have a built in headphone socket, but if it doesn't you will need an external headphone amplifier.
Active monitors are often better value for money in the budget end and more variety has appeared in recent years. There are active monitors for less than £150, such Alesis m1 Active 520 (Tom O Shanter bought a pair). Tannoy Reveal can be got for little over £200 a pair. KRK RP5 or similar might be worth concidering. They are around £280 a pair.
You roughly get what you pay for, but not always.