In Harry Potter, Wizards obviously use some specific transfiguration and conjuration spells, but do they actually need a specific spell for every transmutation and conjuration? Are the specific spells seen in the series simply to make them easier, especially for young witches and wizards?
First, Transfiguration is the general term used for a few different types of spells, anything that changes the makeup of something.
The different types are:
So, transformation changes something, untransfiguration changes it back. Vanishing makes something disappear and conjuring makes something appear.
Now that that's settled, Verbal Spells:
I could not find anywhere in the books that mentioned the names of any transfiguration spells, nor are there any mentions of what they have to say to invoke the spell. The one thing I found is Professor McGonagall saying:
you know the incantation, let me see what you can do
Harry Potter and the Order of the Phoenix, pg. 320
So there must be some type of verbal spell, especially because book 6 talks about practicing Transfiguration without them:
Nonverbal spells were now expected, not only in Defense Against the Dark Arts, but in Charms and Transfiguration too.
Harry Potter and the Half Blood Prince, pg. 217
Everywhere else it just happens with a flash of light and a bang, or no indication at all:
A flash of blue-white light erupted from both wands...
Harry Potter and the Prisoner of Azkaban, ch. 17
... he heard a second loud BANG, and a roar that echoed through the entrance hall.
Harry Potter and the Goblet of Fire, pg. 133
pulling out her wand; a moment later, with a loud snapping noise, Draco Malfoy had reappeared
Harry Potter and the Goblet of Fire, pg. 134
It seems that there would be different spells for each of the above types, along with for the varying complexities of the spells. In the words of Professor McGonagall:
Transfiguration is some of the most complex and dangerous magic
Harry Potter and the Philosopher's Stone, pg. 106
The class Transfiguration is required for the first 5 years at Hogwarts, which would mean there is lots of complexity and stuff to learn. They don't learn Vanishing spells until book 5, again from McGonagall:
...today we are starting Vanishing Spells. These are easier than Conjuring Spells, which you would not usually attempt until N.E.W.T. level, but they are still among the most difficult magic you will be tested on in your O.W.L
Harry Potter and the Order of the Phoenix, pg. 257
The complexity of the spell depends on what you are trying to do:
... the Vanishing Spell becomes more difficult with the complexity of the animal to be vanished. The snail, as an invertebrate, does not present much of a challenge; the mouse, as a mammal offers a much greater one. This is not, therefore, magic you can accomplish with your mind on your dinner.
Harry Potter and the Order of the Phoenix, pg. 320
... submerged themselves in studying for Transfiguration next day. Harry went to bed, his head buzzing with complex spell models and theories.
Harry Potter and the Order of the Phoenix, pg. 714
... the immensely difficult topic of human Transfiguration...
Harry Potter and the Half Blood Prince, pg. 309
When Harry is getting his first wand, Ollivander says:
"Your father, on the other hand, favored a mahogany wand. Eleven inches. Pliable. A little more power and excellent for transfiguration. Well, I say your father favored it -- it's really the wand that chooses the wizard, of course."
Harry Potter and the Philosopher's Stone, pg. 66 (emphasis mine)
This could insinuate that there are other factors that can affect Transfiguration (or Ollivander is full of it, which could easily also be true).
If I'm understanding your question correctly (whether every transfiguration needs a specific spell to be cast by the witch or wizard) I would guess no and point mainly to the few transfigurations that we know Harry himself to have caused before he even knew magic to be real.
One could probably argue that every transfiguration does need a specific spell which would mean that Harry just happened to accidentally perform the non-verbal Color Change charm to turn his teacher's wig blue, as well as a non-verbal Shrinking charm on Dudley's sweater, and a non-verbal Vanishing charm on the glass in the zoo...etc.
That argument seems to imply that there are all these concrete magical spells that float around in a witch/wizard's magical subconscious, available to be accidentally selected for use. I'm more inclined to believe that things in the magical world aren't so concrete and that any particular transfiguration doesn't necessarily have to be limited to a specific spell.
In a way this reminds me of when Ollivander was explaining how "any wizard at all will be able to channel their magic through almost any instrument" [paraphrased]. Just as the physical instrument used to channel a witch or wizard's magic isn't fixed I would say that the spells used to perform transfigurations isn't fixed either.
I think that spell incantations are simply tested and tried methods of performing the same spells repeatedly (magical science?).