Tips? Ok, here goes...
Starting out
Autocannons with High Explosive Ammo for anyone with Strenght over 35. Heck, you could simply go with Autocannons for everyone or fire people with under 30 Strenght. Barring that, Rifles. HE AC is the best weapon available to you early on, letting you autofire grenades like crazy at the enemy.
Remember to get Grenades and Flares. Personally, I don't care much for Tanks, but if you insist, get the Rocket one.
Get Avalance Launchers for your Interceptors. The other two weapons are poop.
People will give you convoluted base building strategies. Don't listen to them, it's not really relevan on lower difficulties and is more trouble than it's worth. When starting out, build Alien Containment, Large Radar System, one more General Storage and Living Quarters. Get some Scientists (10-20) and Soldiers.
Early Research should focus on getting 3 items: Laser Rifles (infinite ammo, autofire, easy to use, good damage), Medikits (save wounded soldiers) and Personal Armor (better chance of staying alive, needs Alien Alloys to research). These will drop your casualty rates drastically and give you a solid chance.
Later on
Make sure you build more bases. You might start as early as during the first month, depending on how much money you're making from selling alien crap. The secondary bases should mainly have a Hangar for an Interceptor, Radar systems and a General Stores. Their purpose is to increase your coverage and make sure your superiors don't get pissy with you.
Try to capture a Navigator as soon as possible. Researching any alien Navigator gets you a superior radar system with 100% detection, long range and details on the enemy craft. The Navigator will usually be in the innermost room of any small UFO, usually armed with a Small Launcher (small, greenish weapon). You can run up to him with a Shock Lance, use a Small Launcher if you researched it, OR (and this is my favourite) park one of your soldiers next to him and get hit with the sleeping gas at point blank range, knocking both your soldier and the alien out.
Get Heavy Plasma. It's the best regular gun in the game, no debate. Don't bother with the smaller Plasma Guns. Also get Blast Launchers, Small Launchers (for capturing live ones) and Plasma Cannons (infinite ammo craft weapons).
For research you want at least 50 scientists once you get rolling. 100 would be a good number to aim for.
If you can capture a telepathic alien (Sectoid Leader or Commander or any Ethereal), research him and the followups until you get Mind Shield, then build one in all your bases. This will reduce the chances of your bases being discovered.
In combat
Reserve units for a Snap Shot. I can't stress this enough. Use this option. Please. It saves lives.
Advance cautiously, prefferably in small groups. Cover all angles and windows (Line of Sight is important, you can right click to turn around without moving). Concentrate your fire on single targets. Use your grenades!
Avoid night missions. Have an idea of how long it'll take your Skyranger to get to the destination and try to arrive during the day. If a Terror mission pops up and you'd arrive at night, wait a few hours. As long as you dispatch a team to the Terror site before it expires (even if it's one second before it expires), the mission will stay active until you get there. If you just HAVE to fight a night mission BRING FLARES AND/OR INCENDIARY AMMO.