Try first without new cables. If it's ok sounding to you, and there's no noise, or hum or whatever, then stick with that.
Figure out how to route the computer soundcard output into the mixer. This way you can use those outputs in a mix as well. You can also choose to record parts into the computer and mix there. But to have everything under faders is nice, and avoids having to align the audio (latency?) later.
Then how to route the monitor output of the mixer to the monitor speakers/amplifier. The monitor output knob on the mixer then becomes a volume control for the speakers, independent of the level on the main output.
What cables do you need? It depends on the connectors on your soundcard. if those are "Jacks" TRS 6,3mm then choose that kind of cable.
here's some information about cables https://en.wikipedia.org/wiki/Balanced_audio
here's detailed information about cable connections Sound System Interconnection
go for cables with Neutrik connectors, don't buy (very expensive) audiophile cables with gold damped connectors those are useless if the other side is not gold either, buy microphone cables (XLR), these are not expensive. It is possible to buy cables with stereo jack plugs 6,3mm Neutrik NC 3 FX - Thomann Nederlandse Cyberstore
best is if you make them yourself, since then you can choose very good cable (Canare, Belden, Mogami etc.) for small amounts of money. something to consider before buying a lot of cables for multi output synths. https://www.youtube.com/watch?v=LvgazaC9t8k
if you don't want to solder you can buy an XLR-stereo jack cable for the main output.
like this the sssnake SMP6BK - Thomann Nederlandse Cyberstore