Well, you asked for my shipping thoughts, and that's what you're getting! Here are some of my favourite Final Fantasy VIII pairings. (And... also one threesome.)
For the benefit of anyone who's unfamiliar with the term (is anyone still unfamiliar with the term these days?), 'shipping' is short for 'relationshipping'. It basically just means finding a hypothetical romantic relationship compelling or interesting. So, for example, if you enjoy the romance between Squall and Rinoa, you could say that you ship Squall and Rinoa, or that you like the Squall/Rinoa ship. If you think a romance between Squall and Seifer would have been more interesting, you're a Squall/Seifer shipper.
Squall/Rinoa and Squall/Seifer are probably the most popular ships in the Final Fantasy VIII fandom, and I can see why! But the ones listed below are my personal favourites.
Squall/Zell
My favourite Final Fantasy VIII ship! The contrast in their personalities is fun, and Zell's very open admiration of Squall is endearing. It's a cute relationship, but the feelings involved are slightly unbalanced in a way I find interesting; I think Zell is always going to love Squall a little more than Squall loves Zell.
I find it interesting that Squall is willing to trust Zell with his ring during the battle with Galbadia Garden:
Zell: Give me your ring, will ya? I just wanna borrow it for a while. I won't lose it or anything. C'mon, what do you say?
Squall: Why do you need it?
Zell: Uh... I can't tell you that. C'mon, just give it to me. You can trust me.
Squall: (What the hell... I like this ring... I... guess it's all right if he just wants to borrow it...)
He wouldn't give it to just anyone! Even if Squall's still very closed-off, he demonstrates real trust in Zell by being willing to lend him something he cares about, without knowing the reason, just because Zell asked.
I think Squall/Zell is actually the first pairing I ever developed a real investment in, from any work of fiction. It's certainly the first pairing I ever wrote romantic fanfiction about. I suspect the scene where Squall saves Zell in D-District was largely responsible.
My first attempt at Squall/Zell fanfiction was written when I was thirteen, and it was dire, but it came from the heart.
Seifer/Zell
Fanfiction won me over to this pairing! I'd never really considered a romance between Seifer and Zell before, but then I read and loved Baconfat's VINDICO arc, which was a great portrayal of the hostility between them evolving into something else.
The VINDICO arc is no longer online, sadly, so I can't push you to read the fics that got me into Seifer/Zell! When I was a teenager, I printed the entire series out - in size eight Times New Roman, so it wouldn't take up too many sheets of paper - and I used to bring it to school with me in a plastic folder, so I could reread it between lessons. In the days before smartphones, you had to be creative if you wanted to read fanfiction when you weren't at home!
I have a weakness for fraught, messy couples who would never actually say 'I love you' to each other, and I'd definitely say Seifer/Zell qualifies.
Zell/Rinoa
This isn't a popular pairing, but Zell and Rinoa would make an adorable couple, and you cannot convince me otherwise.
Final Fantasy VIII focuses very heavily on Squall's relationships with other people, so the inter-party relationships that don't involve Squall can feel a little underexplored. But, from what we see of it, I love Zell and Rinoa's friendship. It's very cute that Zell offers to make Rinoa a replica of Squall's ring, and that he gets embarrassed if Rinoa sees him getting frustrated over the hot dog shortage: 'Huh?! Rinoa's here, too?! Oh man, I'm not always like that, okay?'
For a slightly sad concept: if Squall died, I could see Zell and Rinoa taking some sort of comfort in each other. They would both take Squall's absence very, very hard.
Squall/Zell/Rinoa
To be honest, I think Squall, Zell and Rinoa would make a great romantic trio. I was just talking about how cute Zell/Rinoa would be, and I stand by that! But, because Squall is so important to both Zell and Rinoa, I actually find Zell/Rinoa easier to envision if Squall is involved.
Squall isn't a great conversationalist and dislikes being the centre of attention, which means he struggles when he's alone with another person. Maybe the solution is to introduce a third person to the dynamic. Zell and Rinoa, who are both upbeat and chatty, could keep the conversation flowing easily, taking some of the pressure off Squall.
I genuinely think this relationship could work. These three all get along well and care about each other, and their personalities complement each other nicely. I suspect Squall would occasionally need to lie down in a quiet room to recover, though.
Laguna/Raine
Look, I swear I'm capable of liking relationships that don't involve Zell.
There's just something that appeals to me about Laguna and Raine! I love the way they get along, and the flashback to Laguna's proposal is adorable. It's also nice to see a story of second love; Laguna may have loved Julia before he loved Raine, but that doesn't mean his love for Raine is any less significant.