To record your own music at home and begin establishing a studio you need some fundamental parts to get going. They are:
1. An instrument and/or microphone
2. Something to record into like a basic mixing board
3. An interface from your mixing desk into your computer
4. A computer with sufficient speakers and/or headphones.
5. Some software to record and manipulate your work.
These days you can purchase a combined solution for items 2 and 3 in the form of a computer sound recording interface. This is a device that you can hook an musical instrument or microphone into and the other end is attached to your computer either as a soundcard or via a USB cable. This will work as an audio input and the mixing board part is all covered in the software package.
Depending on your demands this may be the way to go or it may not. If you intend on only putting down one audio part at a time then a computer audio interface makes sense. If you require to record a whole band with a few tracks at once then you'll need a mixing board with adequate channels to deal with this.
One thing you will have to deal with when attaching your interface and recording your instruments and voice is latency. This is the delay between the sound you make and it being played back to you by the computer. Latency can completely throw off your timing if you don't minimise it. A means to achieve this is to use ASIO drivers for your computer interface. You can download free universal ASIO drivers at asio4all.com. Also be sure that your interface includes a preamp.
When you have your audio interface set up and can record and monitor your work without latency you're about good to go. All you need now is some proper computer software to function as a mixing board, sequencer and sampler. Just about every good computer interface will come packaged with some recording software, usually a cut down version of a full product but adequate to get you started.
Otherwise there are numerous low cost and even free programs you can download. There are also numerous communities on the internet where you can share your ideas, get help and support and download royalty free samples to use in your music.
Those really are the bare requirements for a home recording studio setup assuming you want to record your own material (as opposed to just play with loops) and that you already own an instrument and/or mic with the appropriate cables. Aside from these it's also necessary to have some high quality speakers and earphones. These days things like effects and processors can all be found in computer software, although as you advance you may want to acquire some hardware versions of these down the line.
You'll also want to make sure that the acoustics in your recording environment are optimal. Don't forget soundproofing as well. There is a great deal to learn and do to get a good home recording studio setup but in the beginning you can just focus on the essentials. A decent computer, an audio interface and some computer software can start you out very nicely. For more information there are plenty of free guides you can download off the net to get you going.