UFO: Aftermath is a fun RTS that follows in the footsteps of the X-Com series of games. Although it is generally easier and less complicated than the X-Com games were, there are points at which it is possible to screw yourself over and make the game extraordinarily hard to win even if you haven't actually met any game-over conditions. With that in mind, here's some helpful tips for people playing this game for the first time.
1. Changes in UFO: Aftermath are largely trigger-based, not time-based. The difficulty increases when (and only when) you do missions, with plot-important missions seeming to give the biggest boost to difficulty. What this means for you is that you can deliberately delay the biggest advances in enemy strength by putting off doing missions like the Area 51 job, and also by being selective about which missions you take on. The upshot of this is that you can spend your time in the early game capturing a lot of bases before trying to get the plot moving, because the plot will wait patiently for you while you build up your power base. However, you should know that while the difficulty of the game does not increase, neither does the quality of the loot you find in missions.
2. Whenever you capture a base, you get a stash of weapons and armor as well as a facility that you can use to launch attacks, manufacture weapons, or research your enemies. Bases are probably the most critical resource you have, so you should NEVER pass up an opportunity to seize one... unless the base is located in the middle of the Biomass, in which case it will be swallowed as soon as you capture it.
3. Planter UFOs are the most dangerous kind. If you ever see one, save your game and then try to shoot it down. If you fail, give some serious thought to reloading. Why? Because Planter UFOs carry the Biomass, a terrain-altering mutagen that will literally cover the map and swallow your bases. You can repel it with anti-biomass bases once you research them, but if the biomass sprouts up in a bad place where a lot of your bases are (like, say, the middle of the USA), it can destroy a lot of your support structures before you can contain it. You don't need to panic if the Biomass starts growing somewhere far away from you, but you SHOULD panic if you think you're in real danger of losing Europe or North America to it.
4. Initially, the alien enemies you fight won't be too tough, but that can change. Once they start getting sky-armor (blue shields) or bio-armor (green shields) they will become very difficult to kill and also start carrying some very nasty weapons. Late in the game, you will start finding heavy machine guns (collapsible or otherwise), which can kill them more easily, but before that, the most effective means of taking them on is with high explosives.
5. Regarding high explosives, you have an infinite supply of frag grenades from the very start of the game. Don't be afraid to use them. Later on, you'll start finding various kinds of grenade and rocket launchers. Generally, it's possible to carry much more ammo for the grenade launchers, and launched grenades find their targets fairly fast compared to rockets. Rockets travel slightly slower, but do more damage, and tend to hit closer to their targets. For all explosives, I recommend making use of the "Attack Ground" function instead of directly attacking an enemy, partly because you can make better use of indirect splash damage this way, and partly because you, the player, are probably better at guessing where your opponent will be standing in a few seconds than your characters are. Remember, there are NO enemies that are immune to explosions, and a lot of terrain cover can be destroyed with explosions too.
6. Incendiary grenades and acid grenades fill an area with damaging flame/acid. They're great for killing everything in a small room, just so long as you're not standing in the room yourself.
7. The most effective weapons in the game in terms of damage, ammo capacity, and accuracy are the collapsible heavy weapons. Characters need to kneel down to set them up, but they can shoot faster and farther than most other guns, so just a few of them can reliably kill ALL enemies in the game. From the other side of the map. With only a 10% chance to hit.
8. Use weapons appropriate for the terrain you're fighting in. Sniper rifles and explosives are good at medium-long range. Ordinary rifles are good at close-medium range. Shotguns are good at close range. Handguns are awful and should be abandoned in favor of bigger guns at the first opportunity. Heavy machine guns can be used at any range.
9. Remember, you can only have 7 people in your active squad at a time. Therefore, you only ever need equipment for 7 people. Don't stress yourself building 20 armor suits for every character you have.
10. Lastly, remember that you can save your game in the middle of a mission. If you think things are going badly, you can always reload.

Great advice! But I´m still having problem.
(Spoliers!)
I´m currently at the Russian base to get the alien documents (or what ever they are called). The thing is, the aliens all have rocket launchers and kill of my team almost instantly. Should I equip all my soldiers with explosiv weapons or should I use a mix of explosive-and soft-bullet weapons? I am currently not able to manufacture any good weapons so the only things I have are the ones you get by completing base missions.

You're at the point in the game where the important thing is not so much which weapon you use as it is that you hit the alien before he hits you. If you don't have any heavy armor, turret guns, or heavy machine/gatling guns, then my general advice is to use lots and lots of explosives, on the grounds that killing him on the first shot is more likely that way. Grenade launchers work best, in my opinion, mostly because they hit the target quicker, but if you prefer to annihilate the entire area where an alien is located, a rocket-launcher full of phosphorus rockets is good as well.
If you're outdoors, giving everyone some sort of explosive device and trying to kill your enemies with an artillery barrage is great. Sniper rifles are also good. If you're indoors, you might be better served with some kind of medium-range AP-bullet assault rifle, or with phosphorus/acid grenades. In all cases, you want as high a volume of fire focused on a single target as you can manage.
If possible, I'd recommend that everyone carry one long-range and one medium/close range weapon.

Thanks for all the help!
I did finish the russian mission just to get stuck on yet another mission. The one where i´m supposed to save two of the "greys".
Hahaha, sometimes I feel that this games is really cheap when it comes to the difficulty, even on normal.

It sounds a bit like you're rushing through the plot missions. Did I mention in the first post that it's neither necessary nor wise to do that?
By the part of the game you're at, you should be using machine-gun turrets as much as possible. Do you have any?

Yes you did mention it, sorry.
I got 1 Double-Barrel Machine gun and 2 Gatlin guns, should start to use them then.

Yes. Absolutely and without question. The hand-held heavy machine guns are a little slower to fire and not as damaging as the turrets, but they're still badass, especially since they work well even on targets they they have a very low chance of hitting.

Imma gonna just do some shameless self-promotion and practice thread necromancy by tying this to another thread I started:
http://www.gog.com/en/forum/ufo_series/vital_information_for_new_players_of_aftershock

At some point, I (or someone else) should make a similar thread for Afterlight, and we can sticky them all.

You'll find out when you research their armors exactly what the stats are, but alien armors tend to be weakest against hard bullets, and best against alien weapons.