I was dreading the 125 shifters after hearing from them on here. It's really not that hard to do, at least I didn't think so. Maybe I just naturally drive them how they're meant to be driven? I dunno. I'll try and help where I can though.
Firstly I left the kart completely stock on all 3 races, and other than one 2nd-place at GT Arena (came 1st on my second attempt) I won them first time.
Brands Hatch for me was the easiest of the three. You can outbrake the AI ridiculously well into Druids Hairpin, practically brake on the apex and ride the brakes out until the front swings round, then back on the power. Usually you'll get about 3 or 4 places doing this, at least on lap 1. Rest of the lap is fairly straight forward, you don't have to slow down for that left-hander that marks where the GP and Indy tracks go separate ways even though the game tells you to (all the AI do), and only a light dab of brakes round the last corner. You should win fairly comfortably.
Willow Springs was the hardest one for me, you can't go anywhere near the curbs at all or you'll spin or get launched. I can't really offer much advice here since I don't remember much about the lap, sorry

Only thing I can remember is that you can brake really, really late into the left-hander after the long straight that begins the "streets" section. Like ridiculously late, I felt sure I was off straight into the cones but the kart just gripped and made the turn comfortably.
GT Arena seems to be the one that everyone finds the hardest, so I was not looking forward to that one at all. With hindsight it might make sense to shorten the gearing (I was only ever in 4th/5th at the end of the straight), but I managed to win with stock gearing so it's up to you. I made up most of my time between the first and third corners (the two slow right-handers), the AI takes that kink in between fairly slowly for some reason, and you can go flat right through it and brake into the hairpin fairly late. I'll own up and say I did a bit of leaning on the AI during this race, but not ramming them off the road or anything. It's important to keep a tight line round those two double-apex corners towards the end of the lap, and make sure not to bounce over the curbs at all. The last corner you can take flat, you'll start to kick the back end out but it's fine.
Just remember on GT Arena that the AI will start to come back to you in the last 2/3 laps. If you can hold the gap at 4-6 seconds, you should be set.
I've never actually given in-depth advice like this on GTP before, I've always considered myself distinctly average at the game, so I hope this helps even a little bit.