Admittedly, there are a couple major changes that make campaigns much easier. The first is Smash & Grab Missions allowing you to find supplies much earlier. They ALSO don't increase alien activity in the region. TAKE THESE EARLY. The other is increased mission timers allowed me to bring smaller squads that can make completing missions on-time more doable. I seriously think these changes made the game significantly easier. These should either have been offset with a difficulty increase elsewhere, or an explicit disclaimer stating LW2 was made easier. The newest patch additionally seems to do nothing but provide more buffs, so you may need to play on Veteran for the same experience.
General Strategy
Early-game, the mission rewards are pivotal, so I took an approach like "expand so I can find/do lots of missions". I took GTS, comms, then AWC (medic scientist = more combat ready soldiers). My goal was to liberate regions ASAP: supplies rolling in -> upgraded trooper gear -> fewer troopers needed per missions -> more missions. I put Chitin plating on as many soldiers as possible (except snipers) because it meant less down-time/more missions. It also offset not upgrading past Predator Armor until really late. I put off campaign objectives until the end.
My prior campaign failures were largely due to a solider moving to a tile that would reveal a pod at the end of my turn. As such I built around avoiding the need to get close to enemies and having consistent damage.
Tech
I rushed plasma and mostly ignored autopsies, psyionics, and SPARKS. I have a bunch of basic troopers, so might as well make the most of them. If you can keep your weapons a level ahead of where they are supposed to be, the game gets really easy. This is especially true with rangers because you don't need the extra upgrades. It's better to prioritize weapons over armor because if you can wipe pods when they appear, they never shoot at you. You also have to worry less about running out of time.
Builds
Rangers
These are my bread and butter. Between Walk and Fire and Locked On, you can reliably hit enemies most of the time. Rangers also have the best aim progression (next to snipers). Sprinkle in Center Mass, Executioner, Grazing Fire with high-teched weapons + scopes + accuracy ammo... and these guys just become SO reliable. XCOM is balanced so it's more about not fucking up than making amazing plays. On low-level relay destruction, the shotguns are really useful, but otherwise I tend not to use them. My rangers leveled up the fastest, and I made lots of them. By the end of the game I had like 15 max level soldiers. Most of them were rangers. Late game I subbed accuracy ammo for venom rounds.
Shinobis
So many people swear by this class, but I couldn't really make it work for me. They accidentally get revealed a lot and take up squad space they aren't outputting damage for. They also reveal a lot of pods when fighting (this is the opposite of the "not fucking up" strategy). They can make good officers, but I prefer Specialists which I'll detail later. Shinobis did shine more when used as spotting for long range sniper squads, but you don't need a lot of them for this. Again, in theory it seemed nice to always have a shinobi, but I rarely found them carrying their weight. This was especially true after I didn't upgrade their swords for a while.
Specialists
Specialists are great because you get remote hacking + rewards on a good hack. If you take lots of missions, this adds up to lots of extra resources. Early-game I build them all as medics. More soldiers alive + reduced recovery time = getting more missions. Later on I started to have some robot hackers as well. To be honest, by late game the rangers were killing everything so efficiently I almost never used medkits. Because rangers can shoot with a first action, they tend to hang back more. Also because I don't use many Assaults or Shinobis, the Specialists themselves often ended up being the ones in front getting shot, meaning they couldn't be there for others or hacks. Specialists got good benefit from my upgraded rifles.
Over half of my officers were specialists, but because they got hurt a lot, and I take so many missions, I didn't always have time to train officers or AWC until late in the campaign. The are nice to have the Command ability because they don't output much damage. The result is a lower opportunity cost for them giving up an action compared to other classes. Getting bonus XP from kills is nice to offset Officers getting fewer kills in general, so you can keep them leveled.
Grenadiers
This is another favored class that I find "meh". I used them more early-game because of the guaranteed kills. Later on I found them less useful due to the noise of grenades attracting pods, destroying corpses, and weapon upgrades out-scaling grenade damage. Later on I learned how OP they are with incendiary grenades and 100% burn rate (nerfed in latest patch
). I tended to kill everything the turn it's revealed anyway, so I still didn't use Grenadiers a lot. They are slow, so I give them SMG's.I did have a couple support Grenadiers who I used as officers. They still kind of felt like taking up useless space, so I built them as babysitters for rookie-training officers. On missions where I had time to infiltrate a lot of troopers, I'd take the Grenadier/Officer, ~2 elite soldiers, and fill the rest with rookies. Stack the kills on the elite soldiers because the rookies auto-level anyway. It's slow to level this type of officer.
Gunners
Gunners do a lot of damage, and are relatively mobile. They do less damage than Rangers, but it's still sometimes nice to have one as insurance with covering fire if I don't kill everything. Again though, I like to kill everything and find myself more often plain shooting with them instead of using all the perks I buy them.
Sharpshooters
After some experimentation, the only way I build sniper is as high-grounded assassins. They don't do much at low levels, but are worthy later on because they can be reliable with high aim and scopes when leveled up. It's nice not having to take extra armor on them so they can get back-up medkits in case my specialist is down or absent. Having multiple snipers is nice because you don't have to move a lot to kill everything, thus revealing fewer pods, thus fucking up less. They work well with Shinobis in certain missions where you stick them on high ground and blast away. I'd actually rate Sharpshooters up with Rangers if it weren't for the fact that often I can't find good high ground.
I did try a stealth holotarget officer who was ok. Like the Shinobi officer but slightly more useful due to always having something to contribute and not revealing pods in close-bombat. It's not worth the resources to upgrade their guns though, so once revealed it's better to have a Specialist. I also think there's a bug with multi-targetting AoE not working.
Assaults
Seriously, these guys, and Shinobis, are why I lost my prior campaigns due to revealing pods. I want to like them, but they just suck. During an early play-through I did do an entire squad of gung-ho Assaults and Shinobis. It slaughtered everything and worked amazingly well until it didn't. The best thing about Assaults is lightning reflexes for flushing overwatch shots. If you don't build them as storm troopers, then they do everything worse than other classes. The exception is if you don't have psionics and use them as disruptors, but instead of stunning somebody, most other classes can just kill them which is better.
Technicals
These guys are pretty cool. Especially when you give them Tycho from SC2's voice pack. I get stable rockets then all of the flamethrower upgrades. These guys scratch my itch for when I need truly reliable damage without having to reveal pods way up close. They make OK officers because after their rocket and x2 flamethrower shots, they kind of suck, so are good to give orders. It's also sometimes hard to get them up close, which is another reason to have them give orders instead.
Tactics
-Pay attention to solider stats when picking their classes and upgrades. Ex: low aim trooper = technical/grenadier. Low aim sniper = holo-target officer.
-Take your time on missions without timers, and take lots of snipers.
-Hug the edges of the map whenever possible so there are less directions you can trip pods from.
-Kill everything. You maximize XP and won't run into the squad-killing situations where you get caught trying to stealth and instead end up fighting every enemy on the map at once.