First off: I am not an expert. I am new to this, had help in this forum and from an audiophile friend of mine. Still I think there are other newbies interested in this summary of what I found out, but If I misinform on something don’t hesitate to let me know and I’ll correct it.
I have a bass, amp and PC with a DAW and now I want to record my music
That’s nice. First you have to decide on budget and quality requirements. It goes without saying that without budget you won’t get the same quality as you would with. I categorized three possible tears of budget/quality:
- No Budget / Low quality - Fine if you just want to record for practice or send some recorded bass lines to friends and family so they can praise your improvement.
- I have some money (60-150€) - Should be all you need for actually producing songs.
- I want the best possible quality (Money shouldn’t be a question) - Probably more than you need if you don’t professionally create music.
So let’s go through the budget/quality tiers.
I have no money
Option No Money A: Get a 1/4" > 1/8" TRS Adapter or cable. Connect your bass to the sound card line in.
Option No Money B: Play like usual and set up your phone to record. Considering the effort the quality is really good.
Option A little Money: Get an active audio cable. (A/D converter cable - different names, same thing). Should be a bump up in quality compared to the other option.
Option Nice Amp: If your amp got a line out you can connect the amp to the line in. Probably the best quality in the no money option.
I have some money and G.A.S.
Buy a Digital Audio Interface (DAI). There are a lot and I mean A LOT of options out there, but I made a comparison table for the interfaces used by bass players here, because interfaces are not primarily made for instruments and every product series of interface generates a different sound. Make sure you like the sound of the one you want to buy. Trying it out would be the best option, but there are recording youtube videos for almost every model out there.
If you really only want to record bass and nothing else and never want to upgrade your system you can even find cheaper models from most brands like Behringer and FocusRite. All you need is a DAI with an Input.
If you want to upgrade later Phantom Power and how many Inputs and Outputs there are and if it has MIDI I/O might be interesting.
I want to reiterate that point: More money spend for the DAI gives you more options not better quality (in most cases). For example: the sound of the UMC 202HD is most likely exactly the same as the sound of the UMC 404HD.
Depending on your amp you can use the line out or DI to connect to your DAI. If you don’t have those don’t worry - there’s a solution.
Although this post focuses on hardware a quick software mention: the solution is the virtual amp - applied effects in your DAW. The sound coming from the DAI is very clean, but often very sharp and kind of hollow. An amp gives it more body and roundness.
Prices are taken from Thomann (except Zoom, had to use amazon. Thoman doesn’t have it anymore.)
I want the best possible quality That’s not even G.A.S anymore. Too much money?
I won’t go into details here, because a) if you chose this tier you have to inform yourself anyway and b) microphones as a topic are a rabbit hole in itself, but the basic setup for the best quality according to sources like Scott from Scotts Bass Lessons and the youtube reverb channels is to use your DAI together with a mic that records the amp output.
So you plug in your bass to the DAI, route the output of that into your amp and then record the amp output with a mic that’s also plugged into the DAI and then record both Inputs at once.
So what I mentioned before with the software is now done with hardware: Combining the clean DAI sound with the full and round amp sound.
P.S: I don’t know what keeps me from editing the first post in this thread. Would have wanted to put the info directly at the start of the thread 
P.P.S: There is already a thread about DAW (Digital Audio Workstation) - the software part. If you are interested in recording I highly suggest to check it out too: What's your favorite DAW?