The two-phase structure of play is explained well.
Be careful to keep subjects and objects consistent within the rules phrases and sentences. For instance, on page 7, "you cannot end your turn on the same space it began" probably needs an additional phrase making it clear that you're talking about the player's ship. ("You cannot end your turn with your rocket on the same tile where it was at the beginning of the turn.")
You may also be able to compress some phrases to improve readability and clarity. For instance, "Actions completed on Earth" could be referred to as "Earth Actions".
The action explanations are good but need more polish and precision. There's a lot going on in each action, so you need to make sure the terminology is consistent and that the players see the most important information first.
Be careful about introducing irrelevant or distracting information when explaining a rule. For instance, "Scout for Crew" refers to to being about to change the crew member that is face up "without having to recruit them". This is meaningless at this point in the rules, because players reading through the rules will not know what recruiting is yet. It doesn't seem like there is any reason to bring up the difference from recruitment here; you can just specify that scouting is about changing a face-up crew members.
Similarly, "Recruit Crew" starts by specifying the existence of a crew limit. This is useful information, but it's supplemental to the action itself. Start by explaining the most important part of the action ("To recruit a new crew member, take the either the face up crew member..." etc.) and then specify limitations or modifiers in separate paragraphs after you have explained the key idea.
The use verbs of "take" and "draw" should also be consistent. Most people understand that you draw from a deck, but in "Recruit Crew" you are instructed to "take" the face-up card. Is this also a draw? Can you take the face-up card if you already have a crew member of the same title (switching it with the old card), or can you not take that card at all? Most of this question is avoided if you use the verb "draw" throughout the rule.
When doing upgrades (or crew recruitment for that matter), do you replace the face-up card *after* you shuffle the old part into the deck? That's what is implied, but it could be more specific.
"Upgrade Ship" says that only sectors with an outlined square may be upgraded. It would be good to have a visual example here; it doesn't look like this applies to any of the ship parts depicted.
"Reveal a Space" describes the action, then explains that the action "counts as only two actions". You're burying the lede here -- the fact that this costs two actions is the most important part. This is the rare case where you should lead with the limitation, since it is an exception to the overall action structure. Try something like, "Reveal a Space costs two actions. When you take this action, you draw a tile, move to it, and complete the action dictated by that tile."
A quick wording nitpick while revealing spaces: you have "fewer" tiles (countable items) in a pile. You would have "less" ice cream (a measurable quantity) if the game involved scooping out ice cream, which is impractical but might be delicious. But I digress.
Instead of connecting two related but non-sequential instructions with "and", separate the instructions into two sentences. ("Move your rocket to the tile you just placed. You must take the appropriate action..." etc.)
You probably don't need the "Immediately" after each tile, since you are taking the action based on the previous sequence. If you want to stress the immediately, put it in the previous instruction. (For example, "You must immediately take the action associated with this tile".)
The "Space Tile" instruction looks longer and more cumbersome than it is. You might want to set off the example with italics on the "Example:" or giving it its own graphic box.
Space Typo Patrol: In the "Move" action, occupied has two c's.
The "Move" action is confusing. It sounds like 1 Move action moves you 1 space -- if so, this should be made more explicit by replacing the "any" in "any space adjacent". However, it's unclear how you "complete" a Move action that moves you into an occupied space, given the "cannot complete actions" rule. You might be able to sidestep this by telling the players that they "cannot complete non-Move actions" or end their turn in an occupied space.
However, you also have some edge cases that interact with the "must end your turn on a different tile" rule. In theory, I could park myself at a space station, take a couple of actions like searching for parts or recruiting crew that change the information state of the game, and then (with 1 action left) notice that all my adjacent spaces are occupied. How should this be handled?
The "Warp" action. It sounds like moving to the Warp tile is a separate action. The rule makes sense as-is, and accounts for the fact that you might arrive at the Warp at the end of a turn and not be able to use it until your next turn. However, you might want to make the idea that activating the Warp is separate from moving to the Warp a little more explicit, especially since in many cases players will move-and-activate as part of the Reveal a Space action.