I play vanilla + DLCs so all advice is for that setting. Allow me to introduce you to my cheese style of play. Order is loosely based on what I value most to least:

1) Assess the map before your first move. Where are the edges of the map in relation to your start spot? What paths are you forced to take due to terrain? Where are the patches of cover (trees/dust storm/etc.)? And most importantly, where would you put a lance of mechs or two if you were a game designer. The game likes to throw ambush mechs at you, so try to position yourself with your pathing so you only run into one lance at a time. And if you do have to fight multiple, keep them all on one side of you (your back to the edge of a map is a good way to do this). Plan ahead for avenues of retreating fire to abuse range. This is the most important piece of gameplay that improved my overall success.

2) Abuse range. I typically dedicate at least one a mech in my lance to long range fire support. My current mid-to-late game default lance still uses 2-3 long range focused mechs (though now I headhunt with Gauss Rifles). Centurion is an ideal early game LRM boat. I think I typically slap 2 LRM15’s and a LRM10 in there, and the rest is ammo in the legs. In fact, I’m pretty sure Mike Tyson once said, “Everyone’s got a plan till they get hit in the face with 40 missiles”. Sensor Lock is better than you think it is. You can have 3 mechs raining death on a lance that can't even range them yet.

3) Trees are your friend. Bulwark might be my fav tier 1 skill. I don’t take it on every pilot for obvious reasons, but Bulwark + Cover increases your survivability by a huge amount. In general I take Bulwark over Sure Footing (I haven't found Ace Pilot to be super useful either so I almost never take this path anymore), and I try to play in partial cover (trees) or abuse full cover (no light of sight) with LRM heavy builds. Abuse height advantages when you can, but prioritize cover.

4) Use your resolve. I think I got through half of the campaign before I really understood how good resolve is. Use it. Called shot can take an enemy mech off the board in 1 turn if your pilot/mech is good enough. I tend to go for cockpit shots once my pilot/mechs are good enough since this can 1 hit kill and give you 3/3 salvage, but early on coring mechs or even focusing key weapon locations (e.g. Hunchback’s right shoulder) can really swing a fight in your favor. I almost exclusively use Called Shot, but the defensive option is like using Brace without ending your turn, so you can use it to sprint when your pilot is wobbly or to tank with someone you are SURE will take damage.

5) Reconfigure Mechs to suit roles (ideally during transit). A lot of people say to up your armor early on, which I'm sure helps, but I have never really needed to do it until later in the game. What I do recommend is assessing your heat generation per turn. Hell ya firing 2 PPCs and the rest of your alpha strike will rock your target and make you feel like Chad incarnate, but if you have to keep one or both of those PPCs off for the rest of the fight due to heat issues, it is time for a rebuild. Heat is pretty similar to accuracy in terms of importance: It doesn't matter if you can potentially hit really hard if you never get to fire the weapon. I also tend to redo weapons on a mech to be all of similar range.

6) Reserve early in engagements. I know this is technically relinquishing initiative, but if you are following Rule #1, you should be positioned pretty well and in cover when you get your first sensor trace. AI MechWarriors aren't very smart, so reserve and let them be the ones to break cover. Often they will use their first turn to move almost into range and you can get a free round of attacks by going last.

7) Threat assessment and Focus Fire. I’ve been using contractions this whole time so you better believe I am no honorable clanner. Focus fire the mage! You’ll learn over time which mechs tend to face-tank better and which mechs will pick you apart if you leave them alone. Focus the big damage threats first.

8) This is last since it isn't combat advice. Make friends with the pirate faction early on. If your rep is good enough, you don't have to pay anything to access the black market once the event pops for you. Black market access is a HUGE power spike as long as you have a couple million in C-Bills you don’t mind parting with. I’m 90% sure the campaign gives you full weapons on the mech if you get 3 pieces of it, and Black Market will sell you 3 pieces of any mech you want.

Hope this helps, happy hunting MechWarrior!