The Ancient One awakens fast!
Unlike the other phases, only the first player acts in the Mythos Phase (check the first sentence about the Mythos phase in the rulebook page 9)

In your example you'd have 10 turns each before the GOO (Great Old One) awakens.  Some caveats:


10 is just a sample number, various GOO's have between 7 and 14 tokens allowed.  
Not all Mythos cards open a gate (Rumors for example).
There may already be a gate there (no Doom token is placed).
Gates can open via other circumstances (these also generate a Doom token).  


You can never be quite sure how many turns you have left.   That's one of the great things about Arkham Horror.

The terror track is completely different.  The main way that it advances is due to having too many monsters in the outskirts.  It is pretty rare to fill the terror track, but you will see a building or two close fairly often.  

For two players, you are allowed 6 monsters in the outskirts; if there are any more, then dump them and increment the terror track.  (Note that 8 less the number of players is always the maximum number of monsters allowed in the outskirts.)