Here’s a simple path to get started with the vanilla game:
Start a campaign. There will be two short tutorial missions then a more open mission, and then you will end up in control of a mercenary company. You will have several mechs, including a Shadowhawk, a Vindicator and a Blackjack. The Vindy, without any modifications, is a solid choice for short to medium range scrapper and the unmodified Blackjack is a solid medium to long range shooter. Those will be your core mechs. The Shadowhawk stock setup is kinda weird and janky, but don’t modify it until you have enough mechs set up to your liking to fill out a lance of 4 with the Shadowhawk being repaired. For your fourth mech, you are going to have to use one of the lights at first, just pick the least-worst.
You will start in the independent part of the map with a lot of half-skull and one-skull missions. Stick to those. Use your Vindicator as your front mech to get close and your Blackjack to shoot from range. Use your other mechs as appropriate. Try to focus fire and kill one foe at a time. Vehicles are easy to kill, especially if you get close enough to stomp em (Mechs do double melee damage to vehicles on a stomp). Missions where you fight vehicles or take out enemy convoys are a good way to build up some funds. Be careful about the skull ratings and also pay attention to the additional verbiage Darius will give you. The skulls LIE. But if you pay attention to Darius, he will often clue you in, like “we have no intel on this mission” - that’s a clue the skull rating is off. Also pay attention to the money offered for the mission - if it seems high for the number of skulls that means the skull rating is wrong - trust the money, doubt the skull. Even with all that said, Darius is still a dope smoker some of the time. Be aware that on some missions, accomplishing the main goal and then bravely running away is a very viable option.
Do not take on any escort missions or 2 skull missions until you know what you are doing. Shepherding those drunk-driving vehicle dudes is like herding cats, and at the 2 skull and up level, bad things can happen.
As you gain money and parts, start customizing your mechs to your preference. LIke Perky_Goth said, I like to have my mechs focus on one range of shots so I can alpha strike with confidence. I know others like Lego Warrior enjoy a mix of weapon ranges but that’s weird to me.
As an example, when you customize your starting Vindy, strip everything off but medium lasers and SRMs. You’ll notice they have the same (short) range so they make a solid combo. For close range mechs I like to add jump jets. I’d rather have a “slow” 4-movement-pip mech like the Vindy with jumpjets than a fast Cicada with many more movement pips but no tonnage for armor or weapons. By slapping jump jets on the Vindy you make up for a lot of maneuverability and allow it to get close to use its weapons. For that starting Blackjack, I tend to strip out everything and put a couple of AC5s in there - that gives you a solid mid to long range punch without a lot of heat problems.
As you add mechs to your collection, a few to look out for:
I love the basic Centurion, configured as a missile boat. It’s a 50 tonner with only 4 movement pips and as a missile boat I leave the jump jets off. That means it slow but with the long range and indirect fire of LRMs you don’t care. It has 3 missile mounts so you can easily get 35 or 40 LRMs on there - it’ll run a bit hot and be a bit light on armor, but again, missile boat.
The Hunchback 4G is your buddy if you want to rock an AC20 - I usually strip most of the other crap off there and focus on the AC20, plus jump jets to get in position.
The 5-movement-pip 55 ton mechs are all interchangeable IMO but I like the Shadowhawk best due to its high base melee attack of 85. Even without arm mods, a Shadowhawk can punch a light mech into next week. The stock Shadowhawks all have janky configs so you will need to retool them: I tend to go for jump jets, SRMs and Medium Lasers.
There are many other ways to make light and medium lances that are good; I leave that to you.
Generally speaking, the lightest mechs in a tier suck b/c they have the initiative and engine weight of that tier, without the tonnage to really carry it all. As an example, the 55 ton 5-movement mechs are IMO better than the 60 ton 5-movement mechs. By the time you account for the engine, the 60 tonners don’t actually have more usable tonnage than the 55ers and yet they are one whole initiative point less active, and I believe in initiative.
Mechs that I like to customize that you can access fairly easily
Hunchback 4G for ballistic, 4P for laser
Shadowhawk as a puncher
Centurion as a missile boat
Panther as my preferred light mech
Jenner if you have nothing better
Firestarter if you are a weirdo who loves pain
At the heavier tiers, I like:
Thunderbolt: very flexible, can be configged in many useful ways
Catapult: everyone loves missile boats
Jaegermech: weird but the A is a good missile boat and the S is a nasty ballistic platform
Archer: the best missile boat in the game. I love me some Archers
Grasshopper: the best short range/melee light mech killer
Orion: very flexible
Marauder: only available from Heavy Metal DLC and can be hard to acquire but one you get the Baby Jesus will cry. I could explain but really, you need to get a Marauder with a Tactics 9 pilot and experience it.
As you play the campaign, the campaign missions will provide helpful cash boosts, but try to have your lance ready and pay attention to the pre-mission briefings. Many of the campaign missions have special requirements but are very doable with planning.