If I improve an item without having the perk then again after I get the perk (and more skill) will I get to the same result as if I wait to improve it only after the perk?
Yes.
Will the first improvement affect in any way the maximum I can get?
No.
From your answers it seems to me there's no problem if I improve something early on, just that I'll need the ingredients again, which is fine with me.
This is correct. You can improve an item multiple times. The level of improvement available is determined by skill level + perks + buffs, regardless of the state of the item prior to your attempt to improve it. Be aware that in order to improve magical items you must have a specific perk.
Also be aware that you can pimp out your gear (of any material) to maximum level with one smithing perk (steel smithing) combined with 4 pieces of (high level) enchanted smithing gear and a strong smithing potion. You don't have to have elven smithing, for example, in order to improve elven armor and weapons; you only need the perk if you intend to make your own elven weapons and armor.