My fire is a former adept pvp veteran (my myth is also a former lv.14 pvp knight), and back then, immolate was my go-to card. Immolate does a small amount of damage to you, but a larger amount to the target as well, all for 4 pips.
My strategy was to blade, trap, shield, immolate. blading and trapping increase the damage to the opponent, while shielding for fire will help reduce the damage done on you (immolate is also handy for removing traps like curse, hex and fient).
at higher levels, spells like helephant, dragon and efreet are very handy (helephant for pure damage, dragon for DoT, and efreet for damage plus a 90% weakness).
Also, Dont forget about heckhound and link. Heckhound when boosted properly can do a massive amount of damage each round, and link is the only spell in the game that can remove both weakness and infection at the same time, which can be a life saver if you know what I mean.
Your minion will also be a big help. As of this update, he is focusing allot more on reducing the accuracy of your opponent with smoke mantle, and he can also heal you with sprite. He still will use link and fire elf, and the spells he had before the update(not sure about firecat though; I think that spell got taken out).
When you can, use wyldfire to boost the damage of your fire spells by quite a bit.
Steal charm can also help if you use it right, as it can steal 1 positive charm (guiding light, lifeblade, deathblade, etc.) from your opponent for 1 pip, which can reduce the damage their spell does on you.
Just find a strategy that works for you, as well as using decent gear that gives additonal boosts to your school, and with a little practice, I'm sure you can get past private.
Ok well you didn't specify what level you are but I can easily tell you fire is the best 1v1 class out there if not equal to myth. The basis of your strategy should always include your minion, its a crucial part of 1v1 and it serves to take out your enemies shield and now as well can heal, shield, and mantle your opponet. Next would be the Link spell it doesnt cost many pips can take out shields and knock off infects and weaknesses so you should carry lots of these. Always keep a wyldfire up don't let your opponet change it. As for overall deck setup, Basics: tower shields volcanic shields, heals and infection (weakness if you have it). As for being fire you shouldn't be to reliant on blades keep a blade or 2 just in case but most of the time wyldfire is more then enough. Keep meteors to take out your opponets minions and if their minion is shielded and you don't have any other methods to take it out go ahead and use dragon (if you have it). As for actual strategy you can go many ways about this You can charge up for a heckhound (not reccommended if going secnod) and use a link afterwards, you can just continue you to widdle down their health with immolate links and the occasional helephant, wack off shields and helephant etc..
These are just my thoughts on how you can be a great fire as you can trust me the best of the duelist are of that school, hope this helps good luck in pvp!