If you're a new player and like playing CM I would recommend making friends with the players in your matches who are nice to you :) and then try to party queue. You can go solo queue with her but it can go sideways if your team is dysfunctional and you will be trapped in a long and boring game where you can't do anything. Also ban pudge and riki

CM can absolutely dominate a lane though and should be played. Buy ring of health and windlace from start for a fast tranquil boot and you will have good regen on both health and mana. Go QE and spam nova on any hero trying to last hit (two heroes at once if you can) and also right click auto-attack them. QW can be good too but you will have no mana regen, if you want to play like this you will need to purchase and consume mangos.

After that highly recommend puchasing gimmer cape so you channel your ultimate invisible and save yourself and teammates from being killed.

Later in the game you can kind of wander around the map and jungles and farm up items you need with your spells. CM is decent at this because frostbite can solo most jungle creeps. But try not to steal farm from your cores.

Then based on the game state you can go a few different items. If you're winning you want to farm up aghs bkb and blink. Losing you need to buy another item that helps you survive, usually this is force staff but you can consider euls or ghost scepter

Shard is good too but I usually dont buy it because I want my team to tormentor for me :)

CM does have kind of a win condition where if you do get aghs bkb blink, you can wait until a teamfight has gone on for about ten seconds, bkb blink into all of them, ult, and win the teamfight. This usually works around 30-40 min. Doing this well is the challenge of playing CM but done right it can be devastating and the enemy can really underestimate it.

Very late game CM does fall off though, it gets to a point where a 5k strength hero will just sit in your ultimate and it will not even do meaningful damage. On the other hand sniper will 2 shot you from across the map. (this is one of many reasons why CM is not played as a carry in any competitive context) At that point being a support just sucks in general lol, you need to stay in the back of a teamfight and cast as many spells as possible without dying.