As intelligent NPCs, dragons will have a range of personalities and motivations; however, since your trouble is with separating them from ordinary humanoid NPCs, I would recommend playing up the stereotypes a bit to add some distinction.
I will draw mostly from Draconomicon here, since the specified system is 3.5. It has some advice on roleplaying dragons, though a lot more on habitat and things like that.
Firstly, dragons have time. Unless they are acting in immediate defence of their hoard or offspring, there is hardly ever a reason to hurry, and it is better to do something right then to rush. To fill the time between, they will entertain their minds with puzzles, whether benign pursuits or malicious scheming as well as the accumulation of knowledge and treasure.
This also leads to their vanity and arrogance. This might vary a bit more, and some dragons are diplomatic enough to hide this from the creatures they interact with, but when one's life is an order of magnitude longer than a human's, it is easy to consider them lesser. An idea could be to try thinking of ants, or some other insect, compared to yourself. This is probably more prevalent with the ones playing the great game than the copper from the description.
They also will seek fortune and fame from the moment of their hatching until their deaths, unlike most humanoids who would do most of this in the adult/middle aged period of their lives. A dragon's hoard (in terms of monetary value) is it's indicator of status (along with age), and a dragon with a small holding will find itself looked down upon.
Additionally, you can add something more species specific to add a bit of personality between the dragons you do have. I can summarize a few of the canon ones for the types you listed, which you can add to taste.
Copper dragons are pranksters and riddlers, and very appreciative of humour. They are generally good-natured, but can be covetous and miserly, as well as very annoyed with anyone who does not laugh at their jokes.
Green dragons are belligerent masters of intrigue and back-biting. As well as treasure, they have insatiable lust for power and victory.
Red dragons are rapacious, greedy, and vain. So like normal dragons, but taken up to 11. They are the most obsessive treasure collectors, and will know the value and origin of every item in their hoard.
Silver dragons enjoy the company of the lesser races, and will protect those in need, though they are generally fairly hands-off unless there is some genuine need for action. They spend a lot of time in humanoid form, usually as either old men or youthful elves. They are probably closest psychologically to the lesser lived races with regard to time.
I'd suggest using these traits to help distinguish your dragons a bit more, at least until you get to having many dragon NPCs that are interacted with frequently. If you want some additional little things, you could do something like accent your speech (draconic puts the stress at the beginning of words, which could carry over into their Common speech) or adopt a mannerism or two.
For books, you can try Draconomicon, which is where most of the info above is drawn from. The monster manual has a tiny bit, though I'm assuming you've read that. There are a few third party books that also deal with dragons, though they might have some differing views compared to official Wizards products (though as DM, you can change whatever you like). For more general information, I'd look into the characterization of very long lived characters in fiction.
Best Answer
Full Dragon: Not until very high level/with high level friends; probably a poor plan
If your character has access to someone who can cast True Polymorph, the caster can transform your character into a dragon with a CR equal to or less than your level.
If you're casting it yourself as a 17th level Wizard or Bard, that means you have all the Adult Dragons avaliable. If someone else is casting it on you and you're a lower level, Wyrmlings are CR 3-4; Young are 6-9, Adult are 13-17.
Note that this will remove your class and racial features. If you're high leve enough to be able to become an Adult dragon this is normally a poor plan given you can throw around Wish/Miracle/etc and dragons don't (normally) have spellcasting.
Dragon-like: From starting
If you're happy being simply dragon-like not an actual dragon, there are several options:
Dragonborn
Dragonborn (phb p 32) "look very much like dragons standing in humanoid form, though they lack wings or a tail", have a breath weapon attack that they can use, and resistance to an element depending on what type of Dragonborn they are.
If you have Xanathars Guide to Everything, there are some Dragonborn-only Feats that make them more draconic, e.g. become tougher or improve their breath weapon (XgTE p74)
In the Feats for Races Unearthed Arcana (playtest material) there is also a feat that gives a dragonborn wings and a fly speed, but it is playtest material that did not make it into XGtE, so definitely ask your DM before you get set on having it.
Pros:
Cons:
Draconic Bloodline Sorcerer
The Draconic Bloodline Sorcerer makes you better at talking to dragons, better at doing elemental things of your bloodline, gives you a thin layer of scales, and eventually gives you the ability to manifest wings.
Pros:
Cons: