The specifics of your space adventure are covered in the Space Flight document. The short version is that you will need a Shuttle, a Vacuum suit or Reflect suit, and at least 1,500 gold (preferably more) for docking fees. Launch the shuttle and travel around space, destroying 20 enemy vessels to become a space ace. When you are finished, return to Sosaria to continue your quest.
Now that you are a space ace, it's almost time to save a Princess. However, if you've talked to enough tavernkeepers, you know that you must also be at least level 8 for the princess to help you. Levels are gained by passing time, and it's entirely possibly you won't be level 8 yet. In that case, finish your remaining quests and come back later. If you still need to level and there's nothing left to do, try passing repeatedly in any town or castle.
Which Princess you decide to save doesn't matter. To save a princess, you'll need to first kill the Jester, who drops a key. The key may or may not open the Princess's cell—you can check your inventory to find out (you're looking for the key to cell #2). If you get the right key, lead the Princess out of the castle. In addition to the usual rewards for saving a princess, she will tell you that there is a Time Machine to the northwest. Indeed, if you travel to the northwest part of whichever continent you happen to be on, you will now find a Time Machine waiting for you. However, you still need the fourth Gem to operate it.