Try studying the gear specs. Those consoles of old, they were running dedicated chipsets - on-board synthesisers tasked only with sound creation. Hardware back then was very limited, unable to employ multiple tasks by one chip due to lack of memory and proccessing power. There were quite a few famous designs - SID on C64, Pokey on Atari XE/XL, even some kind of FM chip on Sega Genesis. But most of them (SID for example) were simplier, sine-saw-square designs. You can achive that by mixing waveform with noise and converting it all to low frequency 8-bit. That's the synthesis side of the story.
Sound design side is that - software used to compose music with this chips allowed use of various notation effects such as arpeggios, portamentos, vibratos, slides, retriggering and such. It all revolves around how the note is played, not made. To best understand how they did it is to get to know yourself with chip tunes and Trackers. They allow you to trigger sounds with ultra fast (blaster-like) arpeggios, do quick (one-note lenght) slides and stuff like this. All of the trackers - even the advanced ones like renoise - use those pattern commands to get sounds ready.
So it all boils down to two things - study the designs: if you want to make a Sega Genesis sound, you need to go with FM. If you wish to make GB sound, you need to know how to mix (and which) waveforms with noise and what type of degrading apply to it. And so on.
Get familliar with trackers and pattern effects. There's nothing in old videogames you can't do with it. And that's basically all. High pitched square wave with an arpegio of 0xx set to say 023 and decently set decay will produce you some satisfying effects to start with.