Excellent advice so far!
I'll second the advice to keep mobile, especially in the early game. Evasion on light mechs is a powerful thing, and can keep you viable on missions where the numbers say you should be overwhelmed. Jumping behind cover can save a mech from a crippling attack, and combined with cautious use of Reserve you can end up with some powerful strikes on your opponents.
On your first playthrough or two, go with default game settings. They were set there for a balanced game, and were set well in my opinion. Once you've got a feel for the game in default, then you can start tweaking settings for more challenging campaigns and careers. (Higher difficulty gives higher scores in career mode, but are, naturally, harder to complete.)
Don't worry about hiring too many mechwarriors at first. They each cost you on monthly salary, and you only really need 4 at any single job. That said, having a couple backups lets you run more missions before needing to advance the clock to heal injured mechwarriors. Find your own best balance.
Finally, in the campaign the priority missions pay REALLY well, but are usually a little tougher than advertised and advance the plot. They can be a real boon when your mech repairs are about to bankrupt you.
