Can someone give me a rough idea of how I should set up my base and what I should focus on in the first month or two?
Unlike the old games you don't need to worry about constructing a defensible layout. There are really only two main things to consider in a base layout: steam and adjacency bonuses.
Steam tiles are the only tiles you can build Thermo Generators on, which are the most efficient power source available at the start of the game. You'll want to start building toward your steam tiles and constructing Thermo Generators on them ASAP.
Adjacency bonuses are bonuses you get from having multiple facilities of the same type next to one another. Satellite Uplinks, Workshops, and Laboratories all benefit from adjacency. Of those, Satellite Uplink adjacency bonuses are the most important. Plan on building all of your facilities for satellite control in one big contiguous mass from the word go.
Are there basically a few "must have" technologies by a certain time or else it's essentially game over?
Enemy Within kinda complicated this a bit, and I've not got a super-firm grasp on ideal research patterns post-expansion, but a good thing to prioritize is Carapace Armor. It's what makes it possible to consistently keep your soldiers alive in the field, which is how you start building up a squad of high-ranking soldiers, and high-ranking soldiers are how you win fights on the tactical layer in the long run. It can be researched as soon as you've finished researching Alien Materials.
Other important starter things to note: Experimental Warfare research gets you the Foundry, which has projects that let you build upgrades for your existing equipment and also build SHIVs, which resemble but are legally distinct from tanks. Meld Recombination gets you access to MEC troopers. Both of these mechanized units are real useful early game. I usually grab one of them before prioritizing Carapace Armor, and put off the other until after my regular soldiers have some upgrades.
Do I make it a priority to have satellites above all countries asap?
Satellites win games, so yes. Satellites win the strategic layer the way veteran soldiers win the tactical layer. They're pretty much always a priority until you've got full coverage.
I have read that some abilities like HEAT ammo aren't that great anymore. Are the old soldier build from EU outdated?
Most soldier build guides you'll find are still generally applicable; the balance changes to existing skills aren't
too dramatic. HEAT Ammo and Squad Sight are both nerfed but still extremely useful. Robotic enemies are more plentiful and more dangerous in EW than in EU, so even though its damage bonus is cut in half HEAT Ammo is still good, especially since its alternative skill, Rapid Reaction, isn't all that hot with a Heavy's utterly mediocre Aim stat. If you're playing with randomized soldier stats you might prefer Rapid Reaction on a high-Aim Heavy, or if you've got the Slingshot DLC installed and enabled then the special soldier you get in it is a Heavy with abnormally good Aim and he isn't a bad candidate for Rapid Reaction. For the most part HEAT Ammo is still my go-to.
Squad Sight nerf and Snap Shot buff makes a Snap Shot + In The Zone Sniper a lot more appealing than they used to be. I still prefer Squad Sight + Double Tap for the most part, though, even post-nerf.
Any good youtubers out there I should check out to better learn the mechanics?
No idea, sorry.
Also any other tips would be helpful thanks.
Put helmets on all your soldiers to remind them that they are faceless, disposable, and will be sacrificed for the greater good of humanity at your earliest convenience.
I find this helps morale greatly.
