Making it "easier to cast" is relatively straightforward - it reduces the Magicka cost of the spell, meaning you can cast them more often.
This is similar to how things worked in Oblivion. (I'd link you to the UESP page documenting this, but the Site's still getting hit pretty hard from all the Skyrim traffic)
Each spell has a "base cost", which is modified by the following formula:
spell cost = base cost * skill multiplier * perk multiplier * equipment multiplier * dual cast multiplier
skill multiplier = 1 - (skill/400) ^ 0.65
This means that at 100 skill, spells cost 60% of the base cast (further modified by perks and equipment).
This isn't combat related, but as Affe has pointed out below, increasing your magic skills is required to buy high-end spells from vendors.
Absolutely.
Skills
Illusion synergises well with sneak as invisibility does not gaurantee not being detected. The CC spells are also very useful, allowing you to control situations that would otherwise overwhelm your squishiness. Silent casting will be vital, so get it asap.
Destruction will be tricky without deep magicka reserves until you start to enchant your own stuff to reduce cost. But Impact will work beautifully for single target assassinations. With silent casting, I believe you could also use runes for some fun, though on master they will quickly lose effectiveness. I would stick to one element to save perk points (perhaps shock as a useful rarely-resisted element), and don't bother with master level perk. This will be a useful tree against dragons.
Restoration is moderately useful with the respite perk, allowing you - in conjunction with light armour stamina regen bonus - to spam dual wield power attacks all day long. I would not worry about the perks above adapt. Get respite, the magicka regeneration perks, and possibly the "auto heal on zero death" perk that can help out if you are smacked around when dual-wielding.
Light armour is great, and will give you survivability that most mages don't have. It also takes up far fewer perks than Alteration.
I would skip conjuration perks completely, unless you wish to use bound weapons as a role play. You may ultimately wish to level it to 90 at some point in order to provide a non-smithing source of Daedric weapons (see smithing below), and if so don't rush on it until you need to level other skills up to level in general (high 30s onwards). Alteration has few uses in this build.
Enchanting will be useful, allowing you to supplement your lower magicka with regen and reduce costs. Don't bother with -100% cost on any given skill tree, as you will probably only use destruction as a filler against mobs and dragons but if you use it a lot against high level single targets then consider -100%, as the Expert level spells are expensive. I would aim for some +magicka regen (or +magicka - calculate which is more effective given your base magicka) as that can be used for all your magick trees, and you will be using them all not just 95% destruction like in pure mage. Possibly some reduce cost for illusion. All other slots for boosting your melee and resistances. Restoration is fine without reduced cost.
Smithing is great - but you may not feel like role playing it. Depends on your target equipment, but you could make do with steel smithing and the enchanted item smithing perks. You can get daedric weapons through the Mages guild, so don't feel obliged to perk up smithing.
Alchemy complements both skill sets well.
The Beginning
Don't spread yourself too thin to start off with. I would work on illusion, sneak and one-handed until you can comfortably stay hidden in shadows and get silent casting perk. From there you can start on destruction up to Impact, to have a useful fall-back of dual-casting firebolts (it can interrupt dragons and nasty bosses), and get restoration to apprentice level. I would then work on getting illusion to 75 and sneak till 60 for backstab bonuses. By this point you should be able to flit in and out backstabbing and skirmishing, using Invisibility as a means of getting back into cover.
Quests
You will not be locked out of any, and can complete Thieves guild, DB and Mages guild. They all work well together. Thieves guild and DB synergise well, and Mages guild unlocks some tasty equipment.
From here, you can choose. I would work on your sneaky sneaky and melee skills, as you will already have the main utility from your skill sets.
Use illusion to ensure you don't get hammered too much, and don't be afraid to go invis and hide. Your main advantages over a pure stealth character are your much better flexibility when detected - so use these advantages.
Hth =)
EDIT: stats: ymmv but I have had good sucess with balanced melee/mage with 2:1:1 health:stamina:magicka. I leave stamina at 150 (maybe 200 if going dual wield).
EDIT2: added link to find out if +magicka or +magicka regen, as per the paragraph on Enchanting. added link for Mage's guild and daedric weapons
Best Answer
I would work on restoration to 60, to provide additional magicka regen, and cheap Close Wounds. The spells and perks beyond 60 are very much optional.
Alteration is indeed useful, as without damage reduction you will be very squishy at later levels. The entire tree is pretty good, with very useful perks - but it does need a lot of them.
Conjuration is useful as a means of providing a wall between you and enemies. However, it may not be to your playstyle and you can certainly do just fine without it.
Illusion is good. Invisibility is not perfect, and you can still be detected but you can use it to infiltrate at a short to medium range. The CC spells of Illusion require high investment in the relevant perks to remain effective end-game, but you can focus on a particular effect (e.g. calm) to save a few perk points. On the other hand, Aspect Of Terror also boosts your destruction fire damage by 10 points.
I would consider alchemy too, as one of the few ways to reliably boost destruction damage. (glowing mushrooms with nightshade are probably the most common ingredients that offer the effect).
Enchant reduced cost destruction equipment, and use remaining slots to boost magicka regen for use on the other schools. Magic resistance is useful, but with Alteration perks and any base Racial trait (are you a Breton?) one piece should be fine.
As mentioned by others, a character planner can help ensure you have enough perk points. Magic can be perk intensive!
EDIT: Some choices, in response to a comment:
1) A perk planner spreadsheet by Shiloh hosted at SkyrimNexus
2) Graphical Perk Planner by PsychoHampster hosted at SkyrimNexus
3) There are also some excellent online ones, I believe one by IGN as well. But I use offline ones so I don't have links.