I haven't played vanilla in a while, but I remember being able to consistently finish the first mission without casualties. Those linear urban maps are the worse because of lack of destructible cover, but doable with overwatch traps and a bit of RNG aid. Complex maps with buildings and destructible cover are somewhat easy once you get the hang of it because you can abuse the mind merge to kill 2 aliens.

You need to clear the map safely and efficiently. Always be safe, advance slowly getting vision and activating one group at a time, if you are in the middle of the street you don't have to run all the way to the building, take small steps each turn, learn the maps and the places where enemies are more likely to be in the first turns.

When in combat only attack if you have very good chances of killing, if you attack and don't kill they get to shoot you, thus higher chance of you dying. Be good with your grenades, if you want to rely less on luck they are pretty much 80% of your power in the first mission. Grenades make sectoids very vulnerable due to the mind-merge chain kill. After contact move to the best cover possible and if you can't expose at least two of them with a grenade, then hunker down to wait for their mind-merge. Then attack with a grenade to blow up cover and shoot if the other soldiers, netting two kills for the price of one. This is efficient grenade usage.

You have 4 grenades for 8 sectoids, so in the ideal mission you will be using one grenade to kill one pair of mind merged sectoids. If you can't do that, try to at least blow up and damage the two sectoids in the front with a grenade, it is not as good because it will take another shot (and another RNG roll) to kill them both. If you can't even do this you might need to spend one grenade on a single alien. It is very inefficient but you just can't always be efficient on this game. I have won missions where I used 3 grenades on the first pod (I don't recommend it) and missions where I finished without using a single grenade. So sometimes it is ok to use 1 grenade for 1 alien or 2 grenades for 2, specially on the first pods just so you don't activate more aliens. But if you always spend grenades like this you are more likely to fail.

There isn't really much to the first mission except getting vision/cover safely and efficient grenade usage. Yes you may get screwed by RNG, but the odds are not really against you as long as you find efficient plays. The worse comes on month 2, specially council missions. Thin Man are true masters at administering death, you have to kill them in one turn otherwise... well, you know.