As someone who has been roughing it out, here are my tips.
1. Don't be afraid to start over. The game takes a lot of getting used to. You need to learn the proper juggling of building weapons, upgrading your gear, building your base, and supporting the nations. It's a lot more difficult than one might think, and some bad early decisions can really screw you over long term. If things go bad early, with you losing a lot of people early in fights or nations hitting panic mode, don't be ashamed of restarting the game so you can do it right.
2. In combat take it slow. Don't rush. One bad move can make you lose the mission. Take everything one move at a time. Impatience gets you killed. Just last night I was doing a base invade mission, moved my support slightly ahead of the rest of the squad with my last move, and uncovered a group of Crysallids. Using their free turn for movement they ran right at my squad, putting them close enough to attack me on their actual turn and killing two people, making me lose the mission. Never move ahead on your last few moves, don't get overconfident. You never know what is right around the corner.
3, Satellites, Satellites, SATELLITES. Spend most of your money early on uplinks and satellites. They are the single best way to lower panic early on. The goal is to have 4 adjacent uplinks up within a few months so you can cover nearly all the nations. Having satellites to spot UFO activity and Interceptors to shoot them down means that you barely have to learn about panic anymore. They also provide a high income boost, especially if you have Africa's "All In" perk (which is why I recommend starting with your base there). Also, don't place them right away. If possible, save them for the end of the month right before the council report. That way you can save any countries that begin going into panic mode.
4. Don't rush priority research. After you have the Arc Thrower and the Containment slow down and get all your assets up to date. Get up those Satellites, research and build weapons. The enemy will escalate, but so will you. At least get some upgraded armor before moving forward.
Those are the big things I had trouble with in my early playing. I hope it helps.