You had several options, some of which would only be available to you if you held a Swedish passport. If you wanted to travel directly to the UK, there were direct flights from Stockholm to RAF Leuchars, in Scotland, run by the British and the Swedes. The British flights used a variety of aircraft, but the most significant was probably the RAF's Mosquito light bomber, painted in civilian colours and operated by BOAC (British Overseas Airways Corporation). These mainly carried products like ball bearings, but could carry passengers. This was a difficult, dangerous, uncomfortable journey. The Mosquito carried a single passenger, lying on a mattress in the bomb bay, and wearing full flying gear, including oxygen mask, intercom and flying helmet. There was an ever-present risk of fighter attack as the flight path passed over German-occupied Norway. On the 22nd April 1943, one of the BOAC Mosquitos was shot down by an FW-190 as it passed over the Skaggerak, with the aircraft crash-landing in Sweden. The most famous person to use this route was probably the physicist Niels Bohr, smuggled out of Sweden in October 1943. However, the Mosquito was only in service on this route from the last few weeks of 1942, when a single RAF aircraft carried VIPs to Sweden to negotiate a British purchase of Swedish ball bearings. In 1942, the aircraft operating on the route would include the larger, slower Armstrong Whitworth Whitley, Lockheed Hudson and Lockheed Lodestar. The former was an ex-RAF bomber converted to transport, while the others were civilian airliners. All of these aircraft were operated by BOAC. The Swedish operated civilian DC-3 Dakota airliners, operated by the airline AB Aerotransport (ABA). ABA also operated flights to Berlin, to keep Swedish neutrality. ABA flights were safer than BOAC ones, but there was still a risk. In Autumn 1943, several losses to ABA aircraft led to the cancellation of the service until new aircraft were introduced in 1944.
If the air route wasn't attractive, the Royal Navy and SOE operated several blockade runners, running from Stockholm to the UK. These were again intended to bring ball bearings and machinery back to the UK. Most of these ships were Norwegian vessels, interned in Sweden following the German invasion of Norway in 1940. These ships loaded cargoes in Swedish ports, and attempted to run the German blockade to return to the UK for further duty. However, in 1943, five motor-gunboats were converted to fast transports to make the round trip from the UK to Sweden. In 1942, the main blockade-running operation was Operation Performance. Ten Norwegian merchant ships attempted to return to the UK on the 31st April 1942. This was supported, in the North Sea, by a British force of six destroyers and aircraft from Fighter and Coastal Commands. Performance was only partially successful. Two ships, the tankers * BP Newton* and Lind, managed to make it to the UK. Four more ships were engaged by German surface forces, and either sunk or scuttled by their own crews. One was sunk by German aircraft, and another by a mine. The remaining two ships returned to Göteborg to await a future escape.
If you were a Swedish citizen, things became simpler. You could simply travel through German-occupied Europe to Spain or Portugal, where finding passage on a ship or a seat on an airliner to the UK was a lot easier. BOAC continued flights to these destinations throughout the war, despite occasional attacks on their aircraft by German fighters.