Well, in terms of software you need a DAW (digital audio workstation). Although Audacity 'can' be called a DAW, it lacks some midi capability and some other stuff compared to commercial software.. but if you're looking to record your own instruments mix them and add some post-processing it should work just fine.. so in terms of software I'd say it's all you need for this.

Now when it comes to several tracks and high quality recordings your CPU might get a decent hit, so it might be a bit cumbersome to do that on just 'any' old HP laptop, but you can generally manage even if it's slow.. the final export will be what matters anyway. (Common problems will include difficulty playing back in the software after adding effects.. etc, since it's processing the sound in real-time) 

Depending on the mic you use or any electric instruments that can be plugged in, you will probably need a USB audio interface.. the cheapest one, which is very old at this point, but still 'does' the job is probably the lexicon alpha.. you can probably pick a used one up for $10 or so on ebay at this point.. If you get a condenser mic though that will require phantom power which the alpha cannot provide, so in that case either get something a bit more advanced if you plan on recording stuff in the future.. Focusrite makes some really aswesome ones atm. The Focusrite Scarlett Solo is their cheapest / smallest one which should be more than enough for what you need it to do... it outputs really high quality audio and has a 48v phantom power to run condenser mics.. The good thing about these is that you can direct monitor your input and they are also actual external sound-cards so your software can use them for processing sound instead of the onboard one which is usually lower quality (especially in laptops). Now if you only want to record acoustic instruments.. you can also get away with a USB condenser mic, that has it's small 'interface' built in.. a really good one in that category is the Rode NT USB.

Furthermore whenever you record pretty much anything... you need to make sure the room you're recording in doesn't have too much natural reverb, otherwise every single source of sound is going to create a muddy background buzz by reverberating in the room.. As much as a bathroom can sound nice when you sing in it :)) you generally want to be able to control the amount of reverb, hence it's always best to do a clean recording and add the amount you want via software.. not to mention all the frequencies that get altered by bouncing back and forth in the room... So you either need to find some small room where you have a ton of cushioning (carpets.. clothes.. pillows... anything that can absorb sound) or you need to treat your room acoustically..  (You probably saw my videos.. and you noticed I have some acoustic foam behind me... well that's lovely and all, but that's just what you can see on camera.. in reality almost the entire room is covered by either home-made panels or that type of foam - here's a random picture I just took of the corner that's not even near me https://drive.google.com/open?.....lmydU7i0Kw just so you get the idea). You want to create that 'fancy hotel-room' type of 'silence' where your ears can actually feel the lack of reverb. Of course... if you're lucky you might just get away with a random room you have.. in case the reverb there is just fine  

Anyway... hope this helps.. 

Cheers

Ferenc