As the others have pointed out there are lots of ways to skin this cat...
To help clarify and assist you in understanding the different options here is my take on your options
First of all the PC spec looks fine for some recording and @tone has identified your keyboard already has MIDI output so they are 2 good ticks in the boxes
MIDI is a special language that is not "sound" but a set of instructions to (i.e. Which key you play, how hard you hit it, how long you hold it for, whether the sustain pedal was pressed etc.)
This means you can play your keyboard as normal and then record the MIDI data, which can then be used by your PC & software to trigger "virtual" instruments (known as "softsynths" or a "VSTi"), which are essential software based MIDI sound generator modules that translate the MIDI data in to sound - the beauty being you can get lots of different sounds so by playing your keyboard you can trigger a piano, organ, strings, brass, drums etc.
To do this you need the lead suggest by TONE and some software (known as a DAW - Digital Audio Workstation)
In your situation the most efficient way to achieve this (and have software that is used by lots of people on this forum so you will get some help getting it all working 
) is definitely REAPER, which is essentially free until you want to pay for it
REAPER can also record AUDIO - which will cover the singing, and if you want to record the actual sounds from your old synthesiser
To do this properly (and avoid lots of issues) you really need an "Audio Interface" - which is a box of tricks designed to turn the audio in to digital so it can be recorded in the PC and then from digital back to audio so you can hear it play back
The audio interface will also come with special DRIVERS that will allow your computer to process the audio quickly enough so that you can play along with what you have already recorded
I would echo the recommendation @pompeyjazz has made regarding the FOCUSRITE Scarlett range. You may also want to check out the PRESONUS stuff
To record vocals you would also need a microphone and probably some headphones
This would give you everything you need to do what you want
There is a secret option though...
The old romantic in me would love to support your "anniversary project" so if you wanted to record a really rough version of the song you want to record (e.g. on a phone or voice memo app) I would happily "work up" a version of the song for you - but would obviously understand if you want to do it all yourself
This^ is great information, thanks! Do you have any microphones you would recommend?
