Among the other brilliant help offered here is raising the issue of hotkeys.
On pc I do not know about consoles, but they come in handy on a load of situations, even if you rarely change them.
I favour staff, ranged, touch, control, shield, summon, detect, light, heal.
This changes some time dependent on upgrades and enemies, but generally stays the same. Scrolling through the menu gives you a full acess to spells but at the price of combat delay.
If you spread out your hotkey picks you can tap the key enter menu and find close acess to the spell you wish as well.
I've started using a brilliant mod called http://www.tesnexus.com/downloads/file.php?id=30673. It simply adds the ability to set the keys to switch between a set of options and to switch between "wheels" of keys. So I now have my hotkeys set up much the same way I always have (that is, for a pure mage, ranged destruction on 1, touch destruction on 2, controls-- dmoralize, calm turn undead, etc. on 3, summons on 4 and so on) but each key is now a set of choices rather than a single one-- all three elementals on target on 1, all three elementals on touch on 2 and so on. Then I've got a second wheel set up for in town, with charm, fort. mercantile, training spells and the like.
There's another one called http://www.tesnexus.com/downloads/file.php?id=22326 that offers even more options, but it's a more complex system-- much of it is menu driven and you have to hold down a key in order to switch to a different wheel instead of just a single click. It has one feature I really like (a key can be set up as a "gear key"-- a single click switches an entire set of gear all at once, from armor to city clothes, for instance) but since the rest of it is more complicated and I don't really need the rest of the features, I've stayed with STHotkeys.
Back on topic-- I really don't find pure mages all that much tougher to play than any other class, though the strategy, at least in the early game, is much different, and race and birthsign are much more important. I've played fighters, both stealthy and tanky, of virtually any type one might imagine-- from Bosmer pure tanks to Orc light armor archers. There are quirks with each combination, but no real handicaps, since skill and attribute levels tend to smooth out. A character might be a bit handicapped early in the game, but early in the game, the opponents are so weak that it doesn't reallyl matter. And with just a bit of careful planning, the character can be set up so that his skills and attributes are where they need to be by the time they become important, pretty much no matter where they started out. But with a mage, it's different, since so much depends on magicka, and there's a considerable, and fixed, difference between the races and the birthsigns.
To me, the crucial strategy to successfully playing a pure mage is to play it as a pure mage from the very beginning. That's not to say that battlemages, spellswords and the like aren't viable characters-- they certainly are. They're just different-- you can fall back on melee skills and you can take some hits and still survive. If you want to play a pure mage, the most important thing to do is to do so without exception from the very beginning of the game. The reason for that is that it can be done, but it requires a specific set of strategies, and if you don't play the character as a pure mage from the very beginning, you won't learn and practice those strategies and you won't have skill in them when you need it.
For instance-- don't wear armor, ever. Late in the game, you don't want to wear armor anyway, since spell effectiveness becomes an issue with higher level opponents. But that means that early in the game, you don't want to wear it anyway, just so that you force yourself to learn the techniques for surviving while not wearing it.
One of the things I have to retrain myself in every time I switch to a pure mage is constantly moving in fights. Since you're not wearing any armor, you don't want to get hit at all. And fairly regularly, when I switch back to a pure mage, I have to relearn that, and it often happens in the same way-- I'll go after a mudcrab or a rat or something like that, and just stand there like I would with a fighter, and it'll beat the hell out of me. "Oh yeah-- that's right," I'll realize, "I have to dodge." Dodge in-- get off a touch destruction spell-- dodge back out so I don't get hit. Wait for an opening, then dodge in and hit him with another destruction spell, and so on.
You can avoid a lot of that by using on-target spells, but on-target has a much higher magicka cost, so you get less "bang for the buck" and, unless it's area of effect, they can often miss, so you've just wasted all that magicka. On-touch is much cheaper, so you can do much more damage for the same cost, and they're much more dependable. But then you run the risk of getting hit.
Potions are vital. Alchemy's nice, but not necessary-- you
can make do with premade potions. They have the advantage of being instantaneous, rather than working over time (as even the best self-made potions do), but they also tend to be heavier and, since you won't find enough of them just lying around, they'll be an expense. Alchemy is cheaper and more effective (if you have the time to let the potions work) and the potions are generally lighter, so you can carry more of them. But they're also limited-- a potion that gives you 8 magicka a second for 15 seconds isn't going to be terribly useful to you when you're out of magicka and face-to-face with a mountain lion.
Anything that serves as a diversion is great-- conjuration and illusion are the biggies. Conjuring a creature, even if it's not powerful enough to take down the opponent on its own, can often give you the breathing room you need to heal and replenish your magicka. It can also give you a good opportunity to get in a hit or two if you come around behind your opponent while he's fighting the summon. And illusion can give you even more breathing room, by getting your opponents to fight with each other and/or turning yourself invisible.
As Dragatus pointed out though, you really have to stick with illusion for it to pay off, since the best effects aren't even available until you reach journeyman level, and the cast cost is so high that you won't be able to make really good use of most of them even then. There are a few custom spells you can make at a relatively low skill level (very short durations of command, for instance) but the skill doesn't really hit its stride until you're at expert level or so. Conjuration, on the other hand, is useful right from the beginning, and is enormously powerful, if used well, early in the game. Again as Dragatus noted.
Regarding race/sign combinations-- the easiest is Breton/Mage. You get an instant 50% resistance to magic and an extra 100 points of magicka (50 for Breton and 50 for Mage) with no penalties whatsoever. The most powerful is Altmer/Atronach-- that's 250 extra points of magicka (100 for Altmer and 150 for Atronach) but with innate weakness to elemental magic and stunted magicka. Personally, I dislike playing Atronach-- I don't use nuke spells, so the extra magicka isn't that important to me, and the gimmicks I have to use to get around stunted magicka are too irritating to make it worth it to me. That's entirely a personal preference though-- there are many players who won't play anything else, so it all comes down to how you feel about it.
Regarding builds-- the only absolute, as far as I'm concerned, is Magic specialization. I've played everything from the default mage class to a custom build with Destruction as the only magic major (and that one mostly unused) and with a bunch of filler majors for the rest. The only real difference is in how you play the character. With a full set of magic majors, you can pretty much just play the game and skill increases will take care of themselves, though it can make the mid-levels tough, since you'll probably reach those levels before you have the necessary skills to handle things like bear and will o'the wisps easily. With most of the magic skills as minors, leveling isn't an issue at all, but you'll probably have to go out of your way to train the skills, since they'll increase more slowly as minors. Luckily though, training magic skills is just as easy as getting the cheapest possible effect (or, better yet, making a custom spell with the cheapest effect for the shortest duration) and finding a quiet corner and casting it over and over, so, while it might be a bit tedious (particularly at higher skill levels) it's easy enough to do.
It's absolutely imperative that you do the MG recommendations ASAP and get into the Arcane U, so that you can make custom spells. Whatever the school is in which you're specializing, you can almost always get more practical spells by making them yourself. This is especially true of destruction and illusion-- by combining weakness effects and/or other effects and/or using longer durations on destruction, you can make devastating spells that aren't available otherwise, and by shortening the duration of illusion spells, you can get things that are ultimately just as effective as the premade spells and for much lower cast costs.
Really though, the central strategy to playing a pure mage effectively is simply to hold to playing a pure mage. If you successfully fight the urge to give him armor or weapons, you'll have no choice but to learn the ways to work around not having them. The strategies exist-- you just have to stick with it to force yourself to learn them.