Hi, I've been to Jeju a couple of times and usually prefer public transport myself. For trekking, it's generally good, but you'll need to plan a bit.
Buses are quite reliable and cover most parts of the island, including trailheads. From Jeju City, you can catch buses to popular spots like Hallasan National Park (Eoseungsaengak Trail, Seongpanak Trail) or coastal routes. The main intercity bus terminal in Jeju City is a good hub. On weekends, buses are still frequent, but early mornings might have slightly longer waits depending on the specific route. I've found that for getting to a trailhead by, say, 7:30 AM, you'd want to be at the bus stop by 6:30 AM to give yourself a buffer.
For popular trails, the buses are usually direct enough. Look up specific routes on Naver Maps or Kakao Maps – they're indispensable there. Just type in your trailhead, and it will show you bus numbers and times. You can pay with a T-Money card, which is super convenient. While you won't get lost easily, some less popular trailheads might require a transfer or a slightly longer walk from the bus stop.
Renting a car does offer more flexibility for spontaneous detours or if you're hitting multiple, less connected spots. However, if your focus is just one or two well-known trails, public transport is perfectly manageable and quieter. You won't be wasting too much time if you map out your bus connections the night before.