In my experience, here are a few tips that help:
Like it was mentioned before, aliens can see just as well in darkness and light. If you want any chance of success on a night mission, you really should bring your own light. There are two schools of thought here:
1) Use Electro-flares. They are cheap, produce a constant source of light, and you can throw them. (Good when a soldiers' FA sucks).
2) Use incendiary ammunition for light. While more expensive than the Electro-flare, incendiary rounds/rockets can also impart damage on most units. Incendiary rounds are fired/launched from a weapon, good if your soldier' TA sucks.
In early missions, I like the Electro-flare. It only costs $60, and can throw out just as much light as fire. Here is a real hint: if you pick up an Electro-flare before the end of the mission, you recover it as though it was never used! For missions later in the game, I use incendiary rounds almost exclusively for light.
Tanks: I never use them. Pull a tank on to the landing ramp of the Skyranger in a hot landing zone with aliens carrying Heavy Plasmas, and watch what happens. One hit from a Plasma Rifle or Heavy Plasma is enough to destroy this land-based vehicle rather easily. Tanks are also wide targets and have terrible reactions, so you might as well paint a huge bulls-eye on the thing! They don't call tanks bullet-magnets for nothing you know!
I'd much rather lose a $20,000 rookie than a $420,000 tank any day.
Hovertanks? Expensive, but they seem to pull their weight around ok.
Explosives: Thrown (grenades), or fired, these are invaluable against tough aliens. The spillover or splash damage helps to eliminate multiple threats that are clustered together. I like the Proximity Grenade for setting up booby traps or pinning an alien to a spot. The Blaster Launcher is also a good can-opener on UFO hulls, which can catch aliens off-guard and give you a momentary advantage.
Got a hot landing zone? One way to tell is to use a motion scanner first. The other method (which I do not recommend) is to take a peek. If you find many aliens around your ship, use explosives to clear a path, or use fire to create a "kill zone" in front of the ramp. Equip your first troops off a transport with the weapons that have the most damage. The second line of troops off should be the snipers to clean up whatever is left with rifles or pistols.
Armor: Early in the game, you have nothing, so use the terrain to offset this penalty somewhat. Crouching behind trees, hills, fences, and even craft landing gear provide some level of protection. Use smoke grenades to conceal your troop's movement in large open areas.
Terror Sites: Your priorities should be this:
1) Bring your troops back home safely.
2) Kill all the aliens.
3) Keep the civilians safe.
If you know what race of aliens are on a Terror Site, you know what kind of terrorist(s) are there also. Be prepared. Don't just jump into a Terror Site with pistols and a handful of troops and expect to win. Bring as many troops that you are comfortable with, and equip them with the best weapons possible.
Play defensively in early missions. Keep your soldiers safe, and bring them back in one piece. That way they can live to fight another day. Play offensively later on. By this time you should have some armor to let you be more aggressive.
Mission Strategies: Send your troops out in groups of three or so with about 4 tiles between them. This protects them from aliens using grenades or blaster bombs. Do not equip each soldier to handle every possible contingency.
This brings me to another point: Never send the heavy weapon folk somewhere by themselves! One or two soldiers should be his backup in case things get sticky. These soldiers should carry the things that the heavy guy cannot (more ammo, grenades, medikits etc.) and have laser weapons (or plasmas) ready and drawn so they can shoot at a moments notice. Just because a soldier is carrying a weapon with the capability to inflict heavy damage, it does not mean that they can kill every alien that wanders into their line of sight. The soldier might only wound an alien, and everyone knows that a wounded alien is a dangerous animal!
Thoroughly search each and every building you see. Aliens love to hide in buildings on the 2nd floor (that's the first floor for you crazy Brits
) and snipe from windows. To eliminate some of the search tedium, use explosives on the sides of a building to get a good look inside, which prevents a floor-by-floor search.
Stunning: Use the Stun Rod early on. Your troops with the highest Time Units should get the prods because you have to get right up to an alien to use it, and this takes time. The best UFO for stunning is the Large Scout. The bridge is localized and small allowing you to bring multiple units from two directions. You will not get any leaders in a Large Scout, but you probably will get a Navigator on the bridge (research him for the Hyperwave Decoder), and an Engineer in the engine room.
Small Launchers are great for stunning the Leaders and Commanders which you need to finish the game. Your best bet at nabbing a Commander is an alien base. Don't storm the command room upstairs with Small Launchers. Stand below and fire some stun bombs at the floor above first. Now you can bring up another guy with a small launcher to mop up any aliens that are still kicking. In any case, use a Mind Probe to see what you are trying to stun.
Searching for Aliens: Most can be found inside the craft itself. Like I said before, if there are buildings at a landing/crash site, aliens are probably hiding in them. A handful (or more depending on UFO size) are usually dispersed throughout the map on the ground. Try to secure the UFO as soon as possible, and then send the remainder of your troops to search out the alien scouts.
Base Building: Build only one base per continent. One Hyperwave Decoder at each base can detect alien ships in a 2400 nautical mile diameter; that's about the size of an entire continent. Personally, I keep my bases off the Poles. It's expensive to build there, and you really don't get any funding from these areas, so why bother. Protect the counteries (and your funding) first, then protect the globe. If the countries are happy, then so should you!
While these tips are not exactly short, they are simple. You could switch any of these strategies to fit your personal method of play quite easily. The best tip is to have fun and to learn something from every mission. Hope this helps!