There are endless methods for transplanting seedlings or rooted cuttings. They can go into soil, clay pebbles, coco, myriad hydroponics setups... One method that's often overlooked is rockwool.
If you're growing in hydro or soil, transplanting into rockwool may be a good idea. Rockwool comes in blocks that will allow a young plant to get established, or you can use rockwool cubes. The advantage of using rockwool is that it's a clean and recyclable, and it holds moisture evenly, allowing your plants ready access to your nutrients. Plus, it's easier on the planet than peat. Grodan offers a nice tutorial on how to transplant your rooted cuttings into rockwool.