Are you only planning to record bass, or might you expand into more instruments later? How much are you looking to spend?
To start with, a simple audio interface. Look for something by a well known manufacturer that has good reviews. Make sure it has an instrument input so you can connect your bass directly. Something like the
Focusrite 2i2 or
Presonus Audiobox USB is a good place to start for under £100.
Next you'll need a DAW to record and edit audio. I use
Logic but I'm assuming you're on PC so that's not an option.
Reaper is cheap and very good, also worth considering is
Studio One 3 (I think this comes included with the Presonus interface). Most well-known DAWs have a free or entry level version, or at the very least a free trial.
I would stay away from Pro Tools as a beginner simply because it's so complex if you've never used a DAW before. There are much friendlier (not to mention much cheaper) solutions.
Once you have all that installed, all you need to do is plug your bass into the interface, set the gain and record. You'll probably want to find some sort of bass amp simulator to get the tone you want, most DAWs include one nowadays.
That should be all you need to get started. Down the line you'll want to add studio monitors, microphones and maybe some outboard gear but nowadays you can start recording with hardly any tools at all. If you have any questions I'd be glad to help.
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