As the question title says, what's the fastest way to travel in the Harry Potter universe?
In the order of increased fastness:
Definitely not instantaneous, but pretty fast:
'But don't panic and get out too early, wait until you see Fred and George.' (CoS)
Seems to be a couple of seconds:
It happened immediately: Harry felt as though a hook just behind his navel had been suddenly jerked irresistibly forwards. His feet had left the ground; he could feel Ron and Hermione on either side of him, their shoulders banging into his; they were all speeding forwards in a howl of wind and swirling colour; his forefinger was stuck to the boot as though it was pulling him magnetically onwards and then –
His feet slammed into the ground; Ron staggered into him and he fell over; the Portkey hit the ground near his head with a heavy thud. (GoF)
This seems the fastest subjectively, but of course, given overall JKR Maths proficiency, she wasn't as precise as one would have preferred for numerical comparisons:
Harry felt Dumbledore's arm twist away from him and redoubled his grip; the next thing he knew, everything went black; he was being pressed very hard from all directions; he could not breathe, there were iron bands tightening around his chest; his eyeballs were being forced back into his head; his eardrums were being pushed deeper into his skull and then –
He gulped great lungfulls of cold night air and opened his streaming eyes.
Twycross stepped forwards, turned gracefully on the spot with his arms outstretched and vanished in a swirl of robes, reappearing at the back of the Hall. 'Remember the three Ds,' he said, 'and try again... one – two – three –'
Also, there are two more ways to travel near-instantaneously:
Vanishing cabinet. We don't know how fast it transports you or if it's using one of the 3 methods above (no canon info), but presumably it's as fast as the other methods.
As @Kushtrim P. noted in a comment, you can use time-turner+normal travel method (your speed of spacial travel may be as slow as you want, but absolute arrival time is arbitrarily small due to temporal displacement). That is indeed a very well known method of travel in Science Fiction.
There are two distinct methods of teleportation in the Harry Potter universe: apparition, and using a portkey.
A portkey takes longer to setup, but once it's ready, all the traveller needs to do is grab it.
Apparition has no setup time, but does need a spell and concentration.
So depending on whether you want to include the setup time and/or the time needed to cast a spell, it can be either of those.