The first jet aircraft to fly was a German aircraft in 1939. During the war period quite a variety of different designs were constructed by the Germans.
Outside of Deutschland, the first jet to fly was the Gloster E28/39, on 15. May 1941, from Cranwell in Lincolnshire. The first production types, the Gloster Meteor and the De Havilland Vampire first flew in 1943. The Meteor was the only one of the two that actually saw service in the war, most famously attempting to deal with V-1 doodlebugs.
Towards the end of the war, Luftwaffe squadrons had also started to be equipped with jets also, the best known being the Messerschmitt 262, but no direct confrontation between jet aircraft from opposing sides is known to have taken place.
Most people are aware of the way that the victorious powers were able to gain the benefits of German rocket technology after the war. A similar thing happened with German aircraft technology, two developments especially - the swept-back wing and the axial-flow jet engine.
New swept-wing aircraft became known as second-generation jet aircraft, and the way was lead by the Americans and the USSR. The F-86 Saber was originally a traditional straight-wing aircraft in its original design. But when the fruits of German research became available, it was re-designed as a swept-wing aircraft. This caused a slight delay and was a bit of a gamble but by the time of the Korean War, the Saber and the swept-wing Mig-15 were the pre-dominant fighter aircraft. First generation aircraft like the Meteor and Vampire learnt about their inferiority the hard way and were re-assigned to other roles e.g. ground attack.
A good example of the changes introduced during this period was the F-84F Thunderstreak. The original F-84 was a straight-winged first-generation aircraft, called Thunderjet. Nominally, the F-variant was an attempt to upgrade the F-84 with the new technology, although other general changes did produce a greatly different aircraft.
The attempts to produce an all-British second generation aircraft re-kindled the rivalry between Supermarine and Hawker which had existed in the War between their Spitfire and Hurricane respectively. This time the competition was between the Hawker Hunter and the Supermarine Swift. The Hawker was the clear winner.
The first British bombers to incorporate swept-wings were the V-Bombers, the Vulcan and Victor. There was however a first-generation bomber produced, the Canberra which was actually very successful.
The Avro Vulcan and Victor were designed to be fast, high-flying bombers, in contrast to the comparitively slow and low-flying wartime bombers. Part and parcel of this was that they had no defensive armament - they were deemed to be safe from enemy interception by virtue of their speed and height.
Most second-generation fighters were capable of breaking the sound barrier in a dive (and the Victor bomber did once achieve this feat) but the next generation achieved sustained supersonic flight in level flight. The first aircraft to do this was the Super Saber. The first British aircraft to achieve this feat, and actually the only all-British supersonic aircraft ever, was the Lightning.