I used docker for my homeserver for several years, but managing everything with a single docker compose file that I edit over SSH became too tiring, so I moved to kubernetes using k3s. Painless setup, and far easier to control and monitor remotely. The learning curve is there, but I already use kubernetes at work. It’s way easier to setup routing and storage with k3s than juggling volumes was with docker, for starters.
Both are ways to manage containers, and both can use the same container runtime provider, IIRC. They are different in how they manage the containers, with docker/docker-compose being suited for development or one-off services, and kubernetes being more suitable for running and managing a bunch of containers in production, across machines, etc. Think of kubernetes as the pokemon evolution of docker.