I've used destruction extensively on master, here are my pearls of wisdom.

You don't need any other skill trees to use destruction, you don't even have to use robes. Enchanting helps in the sense you can choose what to put your enchantments on, but you can find good enough gear to reduce the destruction costs. Unfortunately once you hit the adept/expert level spells the costs absolutely skyrocket, you have to use gear to reduce them.
You can find/buy:
- Master robes 22% destruction reduction.
- Peerless robes for 25%.
- High level (dragonscale, dragonbone or daedric) armor pieces with peerless destruction 25%.
- Ring for 25%.
- Circlet for 25%.
- High tier helmet for 25%.
Obviously these items are quite rare, especially the armor pieces, but there is a sure method to get daedric armor/helm of peerless destruction, using the atronach forge under the college of winterhold using a sigil stone (90 conjuration needed). Any three of the above will provide 70-75% reduction, which is good enough.
If you plan on enchanting, you don't need to worry about finding the right equipment as you can make your own. Aim for around 90% cost reduction, the perks grant 50% reduction if you need it. With the above robes setup, master robes 22% ring 25% and circlet 25% you get 72% reduction, so you end up with 86% reduction with the perk [100 - 0.5x(100-72)], this is definitely enough. If you enchant you don't need to invest in the perks, but you may struggle at low levels when your enchanting skill is low.
For damage perks unless you have RP reasons (vampire not using fire magic for example) use all three elements, get all 6 augmented perks as soon as possible. The shock disintegrate perk is brilliant, as is the frost paralyse perk, the fire fear perk is worthless. The aspect of fear perk in the illusion tree adds 10 damage (15 if you are a necromage vampire) to all fire spells including flames. The necromage restoration perk increases damage dealt to undead by destruction spells, so when using fire (most undead are weak to fire) you absolutely destroy undead.
Little tip: the frost/shock cloaks are affected by the disintegrate/paralyse perks.
The dual cast + stagger perks are useful, don't dual cast every single attack though. You have 2.2x the damage for 2.8x the cost and 100% chance to stagger with the perk. You don't need this perk though, I tend to just throw a dual cast in every four or five casts.
If you choose to be a necromage vampire (a vampire with the restoration perk necromage), every enchantment/active effect you have is increased by 25% and every spell/potion duration is increased by 50%. Being a necrovamp also increases some perks by 1.25x, the aspect of fire bonus mentioned above is increased from +10 to +15, nearly all illusion perks are increased 1.25x (+8 would now be +10). But you have to take the perks after being a necrovamp.
You should learn what enemies are weak to, trolls are weak to fire, dragons are weak to the opposite element they use, undead (draugr, vamps) are weak to fire for example. Quite a lot of human enemies are nords, so they are 50% resistant to your frost spells, fire or shock is preferred.
Once you have a good setup, you will be dishing out 100 (or 105) damage incinerates, which literally kill draugr deathlords in three dual casts on master with necromage. Bandit marauders (the highest level bandit) have 900 health on master, so you would need 10 expert spells or 9 incinerates with the aspect of fear perk to take one down on master, do remember you can fire those spells incredibly fast, much much faster than shooting a bow.
That's pretty much it, I wouldn't power level any skills as that bores me, level up skils naturally and with the above advice you'll find destruction pretty dam powerful.