The quickest way, I think, to sum up my understanding of why Slovenia was allowed to leave so peacefully is that it effectively had no Serb minority. Zagreb and Belgrade had been moving apart since the 1970's. It wasn't the break up of Yugoslavia that was intrinsically a problem (though for many that was a problem and by the 1980 census something like 10% of the country listed their ethnic identity as just "Yugoslav" and demographers expected that to rise to 20% by 1990--that is, until ethnic tensions began increasing).

But the Serbs had very specific goals when they started fighting in Croatia and Bosnia. They wanted to have a state with all the Yugoslavs Serbs in it. Unfortunately, here's roughly what Yugoslavia looked liked before the war. Notice that Slovenia is basically mono-ethnic Slovenian. Notice also that there are large pockets of Serbs in Croatia and Bosnia, and large pockets of Croats in Bosnia. The Serb nationalists generally wanted to be able to connect up with the Serbs in Croatia, especially a region called the Krajina (most of the inside of the elbow that is Croatia) tried to break away from Croatia as Croatia tried to break away from Yugoslavia. The initial goal for Serbian nationalists was to be able to connect up with this group, who were resisting. This lead to a lot of ethnic cleansing in order to try to create a permanent corridor connecting the two. Slovenia simply didn't have that problem because it didn't have the same ethnic mixture of the other Republics (there are certainly historical reasons for that, but I'm not familiar with them). One of the reasons Bosnia got it so bad was because for a long period it was a three way war, before the Croats and the Bosniak Muslims eventually allied.