Naval rules?

[Doctor Atomic]

I need to move troops across a river, do I have to have a sector along the river in order to build ships? Do I have to have a colony adjacent to the shore to build ships?

 

[sabasNL]

I need to move troops across a river, do I have to have a sector along the river in order to build ships?

You don't need ships to move land units across a river. In fact, ships can't enter a river. Rivers only improve a sector's food production.

You don't need ships to cross water sectors (coasts and oceans) either; most land units will automatically embark on transports while some units are amphibious. You do need to research the proper starter tech before you can embark units, however (though perhaps not for all factions; not sure).

Do I have to have a colony adjacent to the shore to build ships?

Yes, because that allows you to annex a coastal sector to the colony. Annexing one creates a naval base from which you can build ships.

 

[The Founder]

Rivers are terrain features. They usually stop the movement upon entering their square or have a really high movement cost.
The way to get over them effectively is building a road over them. Roads are build between most adjacent colony centers, annexed sectors and - most importantly - frontier outposts. I placed countless numbers of outposts just to get a road connection across poor terrain.

If with river you mean "1 sector wide waterway": You need river exploitation tech to unlock embarking. Embarking and de-embarking costs all remaining movement points, so it is slow.
While there is no mariner skill as far as I can tell, there is plenty of floating and flying units wich just move across as if it was nothing.
Beyond that, there seems no way to speed this up.