Just take a look at the old 80/90s games, and make the game design/mechanics/rules similar. The important thing is to not include any of the mechanics that may have appeared later in that genre (even if they improved the genre), for instance look at the differences between Wolfenstein and Modern Warefare 2 or Gradius and Dodonpachi Dai-Fukkatsu.
And if you want the game to also be retro in aesthetics then you just make the graphics/sounds in a similar style to the old 80/90s games.
The vast majority of games in the 70's 80's and early 90's were either arcade games, arcade conversions or styled on an existing arcade game.
So aiming for gameplay over graphics is the way to go I guess. Fast action is a factor all the good ones had too because arcade games should not bore people.
There did seem to be the idea that every game should give the player 3 lives and have some sort of score to display the player's progress in a game.
Also the better games always had the kind of mechanics that would give up their secrets over time so more experienced players could get further. Say for instance a person playing pacman for a long time will start to notice how the ghosts move and be able to move accordingly whereas a newer player will just charge about and die more often in the process. Something like Bubble Bobble had tons of secrets like the ending digits of the players score determining what powerup will appear in the next screen. Not randomness but something experienced players can take advantage of.
Sound wise? well, I'm sure we all know what arcades sound like.
But really there aren't any particularly consistent styles that made games retro, there have been plenty of different systems made all with their own quirks that changed what the games looked and sounded like. Not to mention the programmers, artists, musicians own styles.
If you really want to go retro, use just 16 colors
Disclaimer: Any sarcasm in my posts will not be mentioned as that would ruin the purpose. It is assumed that the reader is intelligent enough to tell the difference between what is sarcasm and what is not.