As far as I know, choosing to buy the ring leads to the senior route with the blue gauge determining how big a role the older sister plays into the story.
Not buying a ring/introducing to parents leads to the coworker route. The MC's mom wants Sayaka to quit her job to be a housewife and the older coworker becomes her confidant.
Not buying a ring/no introduction leads to the student routes (and the route splits again when you choose who Sayaka attends the event with). Pretty shallow routes IMO.
The father-in-law route was a bit trickier to get as I'm not 100% sure what the priority on it is yet, but it involves cumming inside her and she tells the MC she's pregnant.
There are a few instant game overs that can happen in the game, but they're so abrupt that you can reload the choose the next option without noticing much.
All in all, I was pretty disappointed with this game.
I know a lot of people didn't like Soredemo Tsuma, but I found it pretty nice as things happened over the course of several months. Everything in Marriage Blue feels rushed and the routes themselves were pretty shallow with little to no build up whatsoever. I guess the senior route is the "main NTR" route of the game if you go through the prologue, but even that was facepalm-worthy when the older sister just pops out of nowhere and apparently hates you. The student has a crush on Sayaka so naturally she sleeps with him when he's depressed. I liked how they added a "subheroine route" for variety, but it would help if the girl wasn't a psychopath. The coworker route had some potential, but I skipped through it once the plot disappeared and it turned into a nukige. Speaking of which, I felt the game (outside of the senior route), did little in terms of NTR. The MC literally disappears from the game in a few routes only appearing when some critical choice option pops up. I also hate how they gutted the detective system from Soredemo Tsuma which I felt made it pretty interactive.