Everything levels as you do, but not quite in the way you've described.
If you have an iron sword, it will remain an iron sword with the same base damage for the
remainder of the game. It will do more damage as your stenght is increased and your sword
skill is increased as you level up, but the sword itself will never get better.
However, as you progress through the game better swords types will be available, like
glass weapons or ebony weapons. These weapons and armor types will never be sold in shops
and can only be found in chests or on fallen enemies.
Fur and Iron weapons/armors are available at level 1.
Leather and Steel are available at level 3.
Chainmail and dwarven and available at level 6.
Mithril and Orcish are available at level 10.
Elven and Ebony are available at level 15.
Glass and Daedric are available at level 20.
Although some exceptions might be true, like you might find someone wearing leather armor
before you reach level 3. And you might be able to buy a Steel enchanted piece of armor
before you reach level 3.
From Ebony and on, shopkeepers will not sell these weapons/armors. You'll have to find
them yourself. But they might sell ebony arrows and glass arrows though i'm not sure.
Also, unique weapons and armors level as you do so the longer you wait to get them, the
better they will be. In the case of enchanted items, its enchantments will be much better
at level 20 than level 3. BUT, once you get the item, its strenght and enchantment will
only be as strong as your level.
Ex: chillrend is a shortsword you get as a reward for a quest, if you complete the quest
at level 5 it will be much weaker when you get it than if you completed the quest at level
20. Once you complete the quest even if you level up the sword will not get stronger.
In the case of most questlines, your rewards are leveled, so the reward will be better and
stronger if you wait before completing the quests. Of course, that means running around
and killing wolves and crabs until you reach level 20 which can be quite boring and time
consuming. What i do is i wait until oblivion gates open up all over the place (meaning i
complete the first few quests) and i close them all to level up quickly.
Creatures level up as you do as well, at first you'll notice wolves and trolls but as you
level up, they will be replaced with minotaurs and ogres.
Depending on your level and the difficulty you play the game at, Umbra can be difficult to
deal with. She is one major exception in the weapon/armor level requirements. She has a
full suit of ebony armor regardless of your level, and has a very powerful sword for a low
leveled player. If you know where to find her, and you don't mind setting the difficulty
slider to easy, then you might as well go and kill her right away. Just have a few health
potions and maybe fatigue potions before you take her on.
Also, if you kill her before you reach level 15 her ebony armor will be of orcish
strenght, it will look like any other ebony armor but will not be as strong, still orcish
strenght is pretty good for a low leveled player.