Leroy wrote:

I'm new to the game just started a couple days ago and have a some questions. How long does it take to rank up to the next tier?
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Not much. If you play an hour or two every day, you should easily hit level 50 in under a month by playing all the Featured Episode missions.
Leroy wrote:

How far can a free player get without spending any money? How does one go about obtaining enough currency (dilithium I think?) to buy the next tier of starship?
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Pretty much what the others have said, you get 1 ship token at levels 10, 20, 30 and 40 that you can exchange for a free ship. You also get 1 shuttle token that lets you get a shuttle for free the first time you encounter a shuttle-only mission. These ships and shuttle are more than sufficient for anything the game throws at you (except PvP) if you choose your ship carefully.