I can't remember if it was Melee with my 4 best friends or Soul Calibur 2 with my older brother. I tried to actively improve by just playing against each other constantly and learning their habits and trying to figure out what attacks were the "fastest" or seemed like they had more priority than others. I never attended a local or entered a tournament for either though. It was an effort solely dedicated to being better than my friends/older brother lol. They would talk SO much shit when they'd win, there was no better motivator.
But fighting games didn't really seriously click for me until Super Street Fighter IV. I skipped out on vanilla SF4 because I was intrigued, but not all that interested (and poor lmao). What pulled me in was the combination of Team Spooky streams, the old Fighting Game Weekly heads at GAF including Markman and the IRC channel, and Evo. That created the interest and excitement to get the game. Seth Killian's Domination 101 on the now dead SRK Forums (RIP), tons of up and coming content creators at the time like Vesper Arcade putting out educational videos, again the old heads at the old place, and the entertaining videos like Excellent Adventures all helped me in actually getting better. I remember the (at the time) free Cross Counter Guide to Beating Your Friends pdf was very helpful in enabling me to do exactly that lol.
I discovered how much fun just the process of improving was as well, regardless of the actual match results. Being able to point at specific parts of my gameplay and say with confidence "I've gotten a lot better at this" or eventually being able to look at old replays and being able to see how far I've come were great feelings.
It was a shame though because once I did start to improve by learning outside of our matches together, my friends eventually stopped playing all together. They weren't as interested in the process of improving as I was, which was a really lonely feeling for me. Losses came harder to them, they only wanted to win. But time spent getting better could be instead used for something else, like playing ball. Which I get and is totally fair and valid tbh. I actually intentionally never tried to improve much in Smash in the hopes we could at least continue playing that together.
Although the online play was pretty terrible at the time, I was still too bad myself to have my gameplay impacted in any serious way short of the very worst underwater connections lol. So I had that at least.
Edit: I found out that SuperCombo archives old SRK forum posts for anyone interested in reading Seth Killian's Domination 101:
https://archive.supercombo.gg/c/evo-championship-series/domination-101/8 . Fair warning these articles are absolutely products of their time. Here's a favorite of mine that's basically Seth ranting while still being helpful lol
https://archive.supercombo.gg/t/you-can-lead-a-scrub-to-water-but-you-cant-make-em-think/89175