So I'm going to be taking all of my information for this from Simon Winchester's "Krakatoa: The Day the World Exploded." It's available on Google Books for a look, and will find a copy on library genesis - which I won't link directly to. The entire book seems to be based on this question, so you're in luck if you want some extra reading.
As you might expect, it was vastly different among groups of people. 1883 was in that curious period where science was developing strongly in the west, but people still retained some of their old religious and pagan superstitions. I'm not suggesting they were practicing pagan rituals of course, but if there wasn't a definitive answer for something, they would wonder, and old ideas crept in.
By 1883 telegraph networks were connecting the world. The cables crossed the ocean, and within a short period of the explosion, the news was already being communicated to the other side of the world. Simon Winchester says that "in cities as distant from Java and from each other as Boston and Bombay and Brisbane, all came to know of the event in an instant" and suggests that it was on the day of Krakatoa's eruption and the subsequent telegraphing of the news that "the modern phenomenon known as the global village was born". In essence, it meant that everyone was aware of this one geographical location, Krakatoa. Despite this instant knowledge among operators, it seems the media was slow to publish the news, which led to some of the confusion among the public of what was going on half a world away.
Obviously I can't describe the reaction all around the world, and this would be an enormous post if I did, but I'll bring in a few examples from the book I linked at the start of the post.
In America, the volcanic debris that circled the earth and tinted the sunsets globally was thought to have been enormous forest fires. Firefighters there were scrambled to fight the fires, but discovered nothing there to fight.
Remember that fallback to religious beliefs I mentioned? They knew it was a volcanic explosion, but they couldn't know why. People thought it was an apocalypse, the end of the world. Krakatoa became a "byname for cataclysm, paroxysm, death and disaster" in Victorian England.
These reactions were on the other side of the world though. The sound was heard in Australia and New Zealand, as far inland in the former as Alice Springs. I can't imagine what the reactions were here, but I'll look for more info.
An article I found from 1884 recounts some reactions of people at the time. In the Indian Ocean at the atoll Diego Garcia - which is circled on this map - sailors thought the sound they heard was a vessel in distress. They heard the sound from more than halfway across the Indian Ocean, which is absolutely incredible. Moreover, they didn't hear a faint clap, but rather a bang loud enough to convince them a ship was sinking and needed their attention. Another observation from the article is that changes in wave patterns were noted six days after the explosion. They also mention wave transit, which I think is referring to an ocean wave travelling around the world, and mention that it occurred five times. I could be wrong on my interpretation of that but it sounds nuts.
I'm just looking through some New Zealand newspapers from late August 1883 to see if I can find a reaction from some of the closer colonies. I'll edit them in if I do.
EDIT: New Zealand newspaper accounts added
I've been looking through the New Zealand newspaper archives(I'm not sure what sites are available for other countries, but they're definitely worth checking out if you can find some), and it turns out that no one knew how to spell these funny foreign names back in the day. Krakatoa is variously rendered as 'Cractoa', 'Cracoata', 'Cracatoa' and 'Cracatao'.
In terms of answering this question for reactions though, I may have been wrong in the sound being heard in New Zealand. There isn't a mention of it anywhere I can find in the papers. The earliest report in NZ papers from 1883 on the eruption came on the 27th of August, 1883, in the Nelson Mail. The piece is a tiny 4 line paragraph sandwiched between an advertisement for rat and vermin poison, and a bit of news about Munich railways installing rulers in their tram cars so that children may be measured so only those under a certain height get child fares. It's quite underwhelming for one of the largest volcanic eruptions in modern history. The piece on Krakatoa merely states:
Telegrams from Batavia state that Mount Karang, in the Straits of Sunda, is in full eruption. The shocks are heard several hundred miles away. It is 200 years since the last eruption of this volcano.
According to Google, the distance between Karang and Krakatoa is 74km(46 miles), and one is out at sea. It wouldn't be surprising to guess that the telegram operator had the names confused.
The next reports came on the 28th of August, 2 days later. There were 8 pieces that appeared on the 28th, 7 of which were the same printing of a message from Batavia(modern Jakarta), telling of the eruption.
The eighth piece however, was an eye witness account of the explosion. It's too long for me to copy out, so here's the an upload. The relevant paragraph is the third one down.
