Hey Elle, I would have posted on my previous visit to this thread, however I never played around on my m/s that much. But from mage based experience and also having played a bit of m/s, you basically require anything that any other mage would, as DK said. All the burn buffs he listed, also intensification and magic crossflow which he did not mentioned, should be maxed if possible. As for the arrow skills, wind/fire, I'm pretty sure plussing them doesn't increase the dmg your shot will deal, however you may want to test that for yourself getting fire arrow from 0 to at least 50, because that small amount of TP is an extra 100 crit which is pretty nice. So that covers your 2 skills you use during PvE fights, Flame and Shot, as for PvP aspect, fireball and some may disagree with me here, however fire rose/rise rose explosion. Three reasons why I say these are important; one, fire rose gives a small boost to mdmg, making you hit harder in a pvp match where its much harder to stand and get a flame off instead. Two, fire rose is a dot, dots are unaffected by honor gear, so if you stack your fire rose high, it will be doing pretty good constant dmg to someone who is very tanky in honor gear, also dot's keep rogues out of hide unless escape is used, which even then you can still follow there movement when dmg procs. Lastly, and I think the most important part to fire rose and fire rose explosion, is that m/s' dont have instants past fireball and shot. Now if the person you are fighting is a mage, or a rogue, or a ch/m or ch/w... any class with a hp % absorb bubble, will cause shot to be usless, untill the bubble is removed through time/dmg. Now to take the bubble off a tanky player, will require more then a fireball, which carries a 6second cd. So fire rose and fire rose explosion can be used as a filler between fireball if a player you are fighting is 'bubbled' or evasioned, dodging all your shots. From what i've played around with, fire rose explosion does a pretty decent amount of dmg on top of the hard hitting fire rose dots themselves once stacked.
So to recap, for burns you will need: Flame, Energy Influx, Intensification, Magic Crossflow, Fire knowledge, Wind knowledge, Wisdom all near max if not able to max, with Fire Arrow at around 50-60 for a cheap extra 100+ crit.
For PvP, I'm sure you know most of these however, important mage skills to plus are: Lightening, Silence, Fireball, Electrostatic Charge, Discharge (Not a must however I use often so plussing to 50 does give you a nice little extra amount of dmg at little cost.), Fire Rose, Fire Rose Explosion, Purgatory Fire (however I prefer using T-storm 99% of the time), Thunderstorm.
There are ofc Item-Set skills that are usful in PvP/PvE but I won't get into that as they require no TP. Hope this helps!