I played a Classic Mode game, and lost one country due to really bad timing on swapping out interceptors with firestorms. Other than that, by the time I was hitting Archangel armor, all the countries were in the blue. I completed the game with all countries at level 1 panic (though I think that's actually part of the game).
I'm now in an Impossible Ironman Mode game, with everything I learned from the previous game. Engineers are top priority on missions, unless the panic level of the particular country is 3 or more. (However, I think there's some sort of hidden math behind this - I think engineers can only reduce the build time up to a certain point. I'd really like to give myself like 999 engineers to see if it reduces build time to like 1 day lol.)
Save your satellites. While you may be tempted to deploy them as soon as you have the uplinks available, hold onto them and deploy them to countries that are at 3 or more panic. Deploying a satellite will lower the panic level by 2.
With the above, built sat uplinks as quickly as possible. Workshops, labs, both are secondary to getting the satellite space you require.
Shoot down UFOs, don't let your satellites get shot down! This will instantly put the related country to 5 panic - and if this happens right before the end of the month, you just lost a country's support for good!
Destroying alien bases lowers the panic level of the entire world by one or two points - not sure how many. From what I've noticed, I think alien bases stop popping up once you capture the brainiac at the end of them. In my Classic game, I went almost an entire year after cap'ing that guy at the first alien base I had show up, and never got another base to show up again. Going to test it in my new game by not cap'ing him this time.
In my IIM game, I just researched ARC Throwers, and no countries are even in the yellow right now. Once you learn the tricks, keeping the countries happy is really easy.
Everything endersblade said sums up how to keep the panic down. arc throwers are easy as heck to equip on assault troopers, and you can just run and gun just about everything. I found it was amusing at the beginning to use all my rookies and play it safe with saves, then once I had 4-6 assaults, all arc throwers as spare slots and run and gunned groups of aliens with stun shots. Snipers and assaults are seriously op if you know the tactics.
One thing though, workshops at the beginning are top of the list ender. Reason? you don't have the engineers to keep pumping out those sat ups yet, so those would be the top priority, then sat uplinks (you can pretty much do both). Also, satellites CAN be built ahead of time guys, you just can't launch them unless you have the uplink coverage. I've had three sitting around at once but couldn't fire them off until my uplinks finished coming online. Helps immensely if aliens ever DO happen to knock one out (panic goes to five, pop a sat up immediately, go search for the little bastard who did it, stun the whole crew, snicker to self).
I don't think it's possible to modify panic levels, those seem pretty hardcoded into the game, but I would highly appreciate an option for it, especially since it's permanent when a country withdraws support. Engineers are also high up on the list I would like to see in a trainer as I have been stuck at a low number for months in game now, no mission offering any have come up and I can't progress the story at all since I need a minimum number of Engineers to produce the item required. It just stops me dead and I don't know how long I am going to have to just stop abductions and shoot down aliens until I finally get the additional engineers.