So my original goal was to create a character that uses light armor with a 1h weapon for his right hand and casted destruction magic with the left.
I created a Dark Elf for their bonus to destruction. So in experimenting with this character I would cast a fire spell and it would take several seconds of constant flamethrower to kill it, however when I could smack things twice with a mace and kill it dead much faster.

So, is magic supposed to be weaker in the beginning, or is the game not set up for the type of character I'm playing and my magic will always be gimped?

I find the battlemage/Paladin type characters to be a bit on the weak side now that they can no longer block on demand. The only exception is running with a 1 handed weapon and heal, because you can stand there and tank hits while constantly healing yourself while you normal attack them to death.

When you level up use perks to make it more powerful. I don't use magic so other then that I don't know what to tell you.

Since Morrowind, magic users have always been weaker in the beginning than Meleers. My plan is to reserve it for later in the game.

My issue with magic is I need to hold it down to cast any magic. And I mean, throw stones at me all you like as I compare it to another game, look at a game like Fable 3. You could spellweave, but here was the difference. Press you get a Force Ball or Fire Ball or even a crackle of spark. Hold B and you get a much more refined and powerful spell. I thought in some aspects that is what Skyrim was going to be. Press Right bumper throw a fireball. Hold bumper you get a controlled burst of fire. But the whole entire magic system is based on holding the [censored] thing down and it isn't very fun at all. And Conjuration has been destroyed because of this system of holding things down because guess what...I have to hold to cast my spell. And it should be Press Left Bumper, summon creature. Not Hold and hope to god it works.

In most games magic users start weaker than their melee counterparts but in the end game they are the most powerful characters.

So am I the only one who thinks magic is a lot more powerful than melee? Especially dual-cast.

I found using magic even at the beginning to be more effective than my first character's melee attacks (dual-wielded swords).

Bows do inflict some nice damage as far as I've experienced though, but not as much as magic.

I'm playing a pure mage and I can't say that I'm finding it too weak. Destruction and conjuration are my main skills so far. I like combining spells and firing them in short bursts, it seems to do more damage. When I want to get up close I combine Bound weapon in both hands to create bound dual wielding swords, plus I took the perk which makes bounds weapons do more damage, it rocks.

Yes and no. I like to think of it more as damage over time. Take fire spells for example. You can cast a short blast at an enemy, say 1-2sec for 8-16 damage at novice. However, this also applies a burning effect which increases damage done by fire (maybe per tick, have to read/investigate further.)

So, you can blast an enemy with some fire, set them burning, bash them a few times with a mace while repositioning, then blast them again while they're still burning for increased magic damage.

As someone else said, the perks are really powerful for magic characters. Empowered dual-hand spells, which you can get at level 20 I believe, are really powerful. The first perk in each school reduces the cost of spells in that school level by 50%, and you can get relatively inexpensive "green" items which decrease that by another 15%, and increase magic regen by 50%.

By level 5 I'd say it more than levels out.

Maybe the the game isn't designed to have magic as an off handed weapon. I was going for a warrior mage type character with a sword in one and magic in the other. Doesn't seem to be working out thus far. I'm thinking I should make a pure mage type, or a ranger type character that uses bows and duel wields.

IM AREADY MASTER OF TWO HANDED SWORDS AND ONE HANDED SWORDS! AWESOME GAME !

i'm also playing a pure mage, I find it definetly needs perks to deal with higher level creatures, the 50% less mana perk was a big help, as was dual casting. trying to fight high level enemies without these perks involved a lot of backpedalling, even on the default difficulty (adept i think), afterwards, only high level foes cause any problems.

Another good thing about magic, the flamethrower. Some spells, like the flame spell you start with, can just be cast while bathing a whole group of enemies in flame, taking them all down at once instead of one at a time.

my main difficulty at the moment, magic doesn't seem to have much of a range, i was fighting some archers the other day and was getting pounded by arrows, yet my lightning spell seemed to have no effect until I got much closer (about half the range of their bows). This is annoying as lobbing a fireball at a distant group of enemies used to be an excellent way to open a fight in morrowind and oblivion.

You can still lob a fireball at distant enemies (firebolt)

I was talking about having to hold down the bumpers. It's just not something I'm use to is all.

These hold attack beginner spells are very weak yes, but got special effects, like drain stamina, mana or health damage over time. So you can damage enemies that hold blocking stance, deal extra damage over time between sword attacks, and damage multiple enemies. Though put hardest difficulty or something. Otherwise these special tactics all pointless, if you can kill everything in 2 hits.
Later you will probably just throw fireballs at everyone and get bored

I found different enemies more resistant to certain attacks. And warriors have the disadvantage of being close what makes them pretty vulnerable. Especially when you face an enemy that kills you in 2 hits (restless draugr).

Just wait a bit, you will find more powerful spells and a larger variety of spells, either in tome form or straight from a vendor, early on try going to the court mage in whiterun, he sells some tomes, also look into getting to the college of winterhold, they've got lots of spells there and some free spells too (just make sure you have that novices hood or something that can boost your magic pool if you haven't put points into your magicka, as there is an initiation test that will potentially require you to have a certain amount of magic in order to cast a spell that they may give you).

You will find that the higher level spells tend to be more single cast high damage projectiles that will travel much further than your flames or sparks, also consider putting a perk point into dual casting for destruction, (or any other tree you desire), as that way you will be more powerful at mid to long range.
You could also have a look at some of the other schools of magic, they all have pretty useful spells, (some requiring you to be less direct with your attacks such as the spell, fury, which can (if you feel like sneaking, though thats not strictly necessary) turn your enemies against each other without them even knowing you're there, but even if they do that shouldn't stop you), but destruction is still awesome.