There's a couple beginner guides in the sidebar, I recommend using those. My best suggestion is to use the Trial Stormcrow, stick with your team, know where your weapons fire from so you don't kill your teammates from behind on accident, and get used to using VOIP to call targets if you can. Always lock targets with R, it'll tell you what weapons they have and what shape their mech is in so you know where to shoot.
Smaller mechs tend to run XL engines due to weight concerns, so popping either side torso will kill them (Unless it's a clan mech because fuck them they're special). Generally the heavier the mech the more likely it is to have a standard engine, so popping a side torso will only help disarm them, not kill them.
Know your heat limits. Shutting down in front of an enemy is usually a death sentence (And override shutdown will kill you just as fast, more often than not).
Knowing is like, 90% of the battle. Knowing the maps, knowing the mechs and where to shoot them (Jagermechs, for instance, tend to run little armor in the legs because they're harder to hit and less noticeable than their big, juicy gun arms. So if you're in a smaller mech and go for the legs, they can frequently be an easy kill). But stuff like that... You really only learn it by playing over and over and over. The big thing to learn is where mechs stash their guns, so you know where to shoot. Some variants of mech have no weapons in the left arm, for instance, so shooting that is mostly a waste of time.
Torso-twisting. As you're taking fire, if you're waiting for weapons to cool down, twist a torso into the line of fire. Your arm and side torso will soak damage this way instead of your center torso (And if your center torso goes up, you die). The caveat to this is if you're piloting an Inner Sphere mech with an XL, as if you don't twist your arms in the way you tend to get a side torso focused on, and since those tend to be less well-armored... Well, you tend to die faster.
I highly suggest using throttle decay (check this in your options) because it allows you to accelerate and decelerate faster due to how things are set up right now. Huge difference if you play a style of poking around a corner and shooting.
Some mechs have jumpjets, some don't. If you use them, make sure to reserve ~10-20% of the JJ bar for landing, depending on how far you're falling from. Easy to leg yourself if you're not careful on some mechs that can jump 62m in the air.
Don't buy a mech until you've gotten all 25 of your cadet bonuses (You get one cadet bonus per match, for your first 25 matches). You end up with a nice tidy sum of C-bills that can then be used to buy your first mech, which by then you should have an idea of what to use. I recommend switching trial mechs every 3-5 matches or so to get a feel for what you like.
IF you start liking the game and decide to throw money at it, hands down the best way to do this is to buy MC and use that to buy a Mastery Pack. This gives you three mechs (Which you need three variants of a chassis to master it), as well as 30 days of premium time and the mechbays to house the mechs. Second best way to use MC is buying mechbays. Worst way to spend it is buying individual mechs, because you can pay up to like $20 a mech this way. Buying clan wave packages isn't a bad way to go either, but Wave One requires like $120 to get the first 'good' mech in it, and Wave Two doesn't have a lot of really good mechs... Wave Three looks like it'll be fucking awesome, but that won't be out for a long while yet. The Resistance mechs are pretty rad, and a bit cheaper per chassis than Clan mechs.