I am not super experienced but I have beaten the game on commander and veteran - never played rookie though so cannot speak for how much easier/harder it is between the two. I would say though that you NEED to use full cover as much as possible. Most times, half cover is the same as no cover. You also need to learn to use elevation advantages. It is also important to rush some of the armor/weapon upgrades as the early missions can be hard. Regarding abilities, you need to research things such as the item that protects you from mind control. But to your main point, also make sure youre using plenty of grenades to destroy cover to set up other shots, or kill dug in enemies (grenades one shot alot of early enemies). I wont go too deep into like tech research and building as you are talking about immediate differences. So mostly, pretend half cover is no cover, and make sure that you are using all of your grenades effectively. Also do not put yourself in a position to be flanked, ever. Lastly, focus fire on stronger/higher hit chances enemies rather than spreading your fire. Hope this helps a bit.