The Italian public had never been behind the war a hundred percent. The Second World War was far less popular among the Italian lower classes and the Bourgeoisie than the First World War had been; and the First World War wasn't all that popular in Italy either, so it should give you an idea about Italian support for the war. There was a momentary surge of support after the German success in Poland, Scandinavia, and France, not necessarily because the Italians themselves had contributed to these victories but because with all of the allies dropping like flies it looked as if Italy would finally achieve the dominance of the Mediterranean that the Italian had been hoping for since the Italian unification. The Italians "Mare Nostrum" was within sight.
However, that momentary surge of popularity ended pretty quickly. When the Italian offensive into Egypt was completely routed and the British counterattack pushed deep into Libya. The Italians East African empire fell quickly, the debacle that was the Greek campaign. All these failures sapped what little support their was for the war. The Italians had no great victory to rally around, and when the allies invaded Sicily, the Italians lost any motivation to fight, and Mussolini was pushed out by his own fascist council. Fascist propaganda attempts failed to remedy this. When the allies bombed Sardinia, the fascist newspapers demanded revenge on the American "gangsters" but this failed to elicit a positive response. In fact the Italian people blamed the government for allowing the bombings to happen, rather than blame that actual countries doing the bombings.
The general feeling among the Italian public was one of indifference when they heard of the surrender. People wanted to simply rebuild their homes, bury their dead, and move on. But then the Nazis invaded and seized Italy and disarmed the Italian military. The Italians, especially the soldiers resented the Nazi occupation and the German cause was only hurt when they trotted out the despised fascist leader Mussolini to lead an obvious puppet government. In the South of Italy the allies, especially the Americans, were warmly greeted. This is due to the fact that the South of Italy had never been a supporter of the fascist movement or Benito Mussolini; and many people in Sicily and the South of Italy had family or relatives in America and so they weren't fooled by Fascist propaganda. Most recruits for the Italian co-belligerent Army (the allied aligned army) came from the south, but they didn't get many recruits as there was very little desire to fight another war.
The North of Italy, which was occupied by the Nazis had it worse. Many of the disarmed soldiers and many disgruntled Italian citizens took up arms and became partisans. The communists who had been suppressed by Mussolini also began to fight against the Germans and to carry out reprisals against former fascists. So Northern Italy after the Italian surrender and in the immediate post war period was a bastion for political violence and infighting. So while the South of Italy was happy to be able to start rebuilding, the North had a new set of problems in the German occupation and roving partisans. So while the Italians in the south could start to rebuild and return to a normal life, the Italian public in the North was forced to continue fighting. The major reaction of the public was that they were glad to be rid of Mussolini, who by 1943 was seen as an incompetent; beyond that it depended on whether you lived in the South or the North. If you lived in the South your war was over, those that lived in the North saw that a new war was just starting.
Sources:
Hitler's Italian Allies Royal Armed Forces, Fascist Regime, and the War of 1940–43 MacGregor Knox
The Fall of Mussolini, Italy ,the Italians and the Second World War Philip Morgan
This is the rough outline of how the military, Malitia and rebellions "took the news", with some pre-surrender information for context. Another poster will need to chime in on the population as a whole.
After the collapse of the African Front in 1942, the fall of Sicily and some successful allied air raids on Rome (specifically military targets) in 1943 the popular support for the war effort in Italy had fallen particularly in the southern areas of Italy. As a result of his the Grand Council limited (effectively stripped) Mussolini of power with a vote of no confidence (known as 25 Lugilo) eventually leading to the arrest of Mussolini.
In September of that year, the Allied forces landed to an unconditional and instant surrender from the Italian troops (as in actually went to Docks, instead of beaching and rushing the land) in the far south of the country. This was 5 days prior to an official armistice, but on the same day that the armistice was actually signed in secrecy.
While a new government was set up, Germay provided significant reinforcements (more than tripling their force in the North) and took critical resources. When the surrender was officially announced the German forces essentially changed from "supporting" to "invasion" force (Operation Achse), this also extended into surrounding countries. Some Italian forces resisted the Germans, but that resistance lasted a matter of days.
The Italian people didn't want a fight, the Italian soldiers generally didn't want a fight. Rome was abandoned (partially in order to secure the Kind Emmanuel & the Royal Family). A secret order had been set up for the Italian people to rise against an invading force, it wasn't given until a point where it was useless. Some german malitia, citizens, military continued to resist. Some quite effectively, most notably in out-lying islands like Cephallonia, where the Germans ultimately won and then massacred 5k soldiers - this was the setting of Captain Corelli's Mandolin. These massacres happened elsewhere, this one was the most famous in the Italian theatre, and one of the worst committed by the Nazi forces period.
Italian troops were given a choice when surrendering to Germany, of the 800,000 or so that were captured, almost 90% elected to go into slave labour instead of continue fighting. Another example that Italy and it's Men didn't want to continue.
As well as troops being sent to labour camps, Italian Jews were sent to execution camps (distinct from concentration camps in that they exist with the only purpose of genocide, and not working people until execution). In Japan, Italians were also rounded up and told to choose sides. If you were loyal to The Crown, you were sent to a concentration camp.
Napolise people actively rebelled against the invading German forces before the Allied forces has made it to them. The country was split into a Civil war.