From what I understand, the fundamentals are: charms add properties to an object (like invisibility, buoyancy, etc.) and transfiguration changes what something is (like the substance and shape). So would one use a charm, transfiguration, or a transfiguration charm (if that's a thing)? And what if I wanted to transfigure an object and then animate it, say if I wanted to turn a brick into a toy and then make it fly?
Charming an origami bird, or to use a more HP example, a chocolate frog, to move around almost certainly involves no transfiguration, only charms.
(This is especially obvious with chocolate frogs, since transfiguration can't create food - this is one of Gamp's Laws.)
In the second case, you would use transfiguration on the object to change it, then an animation charm on the resulting transfigured object.
Transfiguration charms (or something close to that) might be a thing, such as with the Weasley twins' joke wands (which, as I understand it, are charmed to transfigure themselves). But your example isn't likely to involve any.
Probably a charm
As mentioned in the question, transfiguration does usually physically alter objects, whereas charms can have a variety of effects.
One of the most prominent examples of animating objects is a charm—Wingardium Leviosa:
On Halloween morning they woke to the delicious smell of baking pumpkin wafting through the corridors. Even better, Professor Flitwick announced in Charms that he thought they were ready to start making objects fly, something they had all been dying to try since they'd seen him make Neville's toad zoom around the classroom. Professor Flitwick put the class into pairs to practice. Harry's partner was Seamus Finnigan (which was a relief, because Neville had been trying to catch his eye). Ron, however, was to be working with Hermione Granger. It was hard to tell whether Ron or Hermione was angrier about this. She hadn't spoken to either of them since the day Harry's broomstick had arrived.
—Harry Potter and the Sorcerer's Stone
This also shows that making objects fly is generally a charm.
To clinch it, in Harry's first year, Professor Flitwick (the charms teacher) had his students animate a pineapple as part of their practical exams.
They had practical exams as well. Professor Flitwick called them one by one into his class to see if they could make a pineapple tapdance across a desk.
—Harry Potter and the Sorcerer's Stone
A combination of transfiguration and animation would then most likely require both a charm and a transfiguration spell.
If you are a hard-working wizard and take your studies seriously, then you will learn to use advanced Transfiguration to turn inanimate objects to animals.
See Harry Potter and the Philosopher's Stone chapter 8.
Professor McGonagall was again diffrent. Harry had been quite right to think she wasn't a teacher to cross. Strict and clever, she gave them a talking-to the moment they had sat down in her first class.
‘Transfiguration is some of the most complex and dangerous magic you will learn at Hogwarts,’ she said. ‘Anyone messing around in my class will leave and not come back. You have been warned.’
Then she changed her desk into a pig and back again. They were all very impressed and couldn't wait to get started, but soon realized they weren't going to be changing furniture into animals for a long time. […]
See also Prisoner of Azkaban chapter 16.
Exam week began and an unnatural hush fell over the castle. The third-years emerged from Transfiguration at lunch-time on Monday limp and ashen-faced, comparing results and bemoaning the difficulty of the tasks they had been set, which had included turning a teapot into a tortoise. Hermione irritated the rest by fussing about how her tortoise had looked more like a turtle, which was the least of everyone else's worries.
‘Mine still had a sprout for a tail, what a nightmare …’
‘Were the tortoises supposed to breathe steam?’
‘It still had a willow-patterned shell, d'you think that'll count against me?’