Some things are good to work on in parallel. And some people have an easier time with certain areas than others.
I recommend learning more about the game, because with game-knowledge, you can take advantage of champions and you won't be caught by surprise by your enemies doing something you don't know about.
Gaining at least a limited understanding of how each role/class of champions works and its strengths and weaknesses can help you to find opportunities to help your teammates along or take advantage of a weakness the enemy has.
Mechanical skill mostly involves positioning and farming. Improving these areas will have a lot of impact on the quality of your play. Being able to position safely or aggressively, manipulate zones of control, juke skillshots, and farm effectively will all help with this.
Teamwork comes down to positioning for each role (such as in a teamfight or against certain classes of champions), and being able to work with people despite being frustrated or having people act terribly towards you or one another. (You can mute chat and pings for specific players by holding <TAB> and clicking the buttons next to their name on the scoreboard.)
I highly suggest watching guides on youtube or reading them on guide websites. You can learn a lot of strategies and skills that you wouldn't otherwise pick up for a long time. Some youtubers who I've learned from include:
- Gaming Curios
- Midbeast (primarily mid-lane)
- Phylol
- Stonewall008 (primarily jungle)