This is a very very long process but First depending on your level you will be limited spell wise but these are how you should spend your training points, the order you use your points for each school is up to you.
quest to level up and train all of your school spells from your teacher, quests, crafting and other trainers located throughout the spiral
Then do the following,
Train to tower shield, this spell is essential to victory in the arena as it defends against all schools by 50%. It is the most used spell in pvp and for good reason. Also train stun shield.
Train to satyr for healing
Train to spirit blade or elemental blade, and weakness
Train to feint or if you want to go further train to doom and gloom.
Train to Thermic shield
Train to ether shield
Train to glacial shield
Train to conviction, fortify, infallible, your school Azteca star spell like cycle of life for , etc. and vengeance
Train to colossal, very important for powering up spells
Train , only your specific school attack spell from khrysalis
Finally train and get shrike if you have to spend a point.
While questing once you get to level 60 farm the waterworks for waterworks gear set once you have that level to 90 and farm Tartarus for hades gear, then finally level to 100 and farm darkmoor for darkmoor gear set, then level to 110 and get the terrors hoard pack level 110 wand that is the strongest wand in the game.
Also train a pet to mega and make sure it has battle talents like pain giver, universal resist talents for 5 and 10% as well as may cast fairy, spritely, or unicorn. The talents your pet has is up to you but this will be the hardest part of your journey to the arena.
Get pierce and crit gems if you can
After you have ALL of that you are ready to fight in pvp now make a general tactic for winning the duel. For example balance spams lore master and gaze of fate to kill or life spams weaver and hungry Catarpillar, etc. you get the picture. Good luck in pvp.
The must have spell in this game is tower shield. If you want to PvP, you need this. This is the most reliable way of stopping a huge spell from destroying your health bar. Getting the Storm + Fire shield as you do it is a boost too, as these two schools hit the hardest.
Weakness is also a strong spell to learn for fire and their overtimes, but not as necessary as tower shield. The other useful balance spell is Triple Blades. If your game plan involves stacking some big hits, this will help you.
Training Life is also important (unless Death) so you can heal yourself in fights. If you really want to be able to heal yourself, get a Life Mastery amulet. These can really boost your survivability.
In Death, you can go for Feint. It's a risky spell, as it clearly states your intentions and can be shielded, but if used well can generate absurd damage. You can also use Doom and Gloom if you want to force a game with little healing.
Astral spells, especially Colossal, are very important as well. Colossal and Auras are key to helping you attain victory. Sharpened Blade further on also helps stack blades if that's what you want to do.
You don't NEED all these spells (well, you NEED Tower Shield) but they help. It's mostly about how you want to play it out.