Does sliding/shifting have to happen in a straight line, or can it be curved?
Unless otherwise specified, movement does not have to be in a straight line. Special movement modes follow the rules for normal movement, except where they specifically differ.
Shift and slide don't specify any extra limitations on the direction of movement, so we follow the default movement rules. If we wanted to, we could slide a creature back and forth one square at a time!
From these rules, pages 9-11:

MOVING
A creature can move a number of squares up to its Speed, entering squares one at a time.
Each new square a creature enters must be adjacent to the last, including diagonally adjacent. However, a creature cannot move diagonally around the corner of a dungeon wall (see “Terrain and Movement” below).
A creature can always move fewer squares than specified by an action or effect (including special movement modes).



SPECIAL MOVEMENT
Some creatures have special forms of movement that affect how they interact with terrain and with other creatures. As well, effects from Order cards might grant special movement to creatures.
SHIFT: When a creature shifts, it ignores the requirement to stop when entering a square adjacent to an enemy creature. If it starts its movement adjacent to an enemy, it can move up to the number of squares stated by the effect (however, its Speed is still 1 for the purpose of effects that state “shift its Speed”). It is still affected by terrain while moving.
SLIDE: Order cards can let you slide other creatures. Sliding a creature lets you move it regardless of who controls that creature.
A sliding creature doesn’t have to stop when entering a square adjacent to or occupied by an enemy and can slide the full stated distance even if it starts adjacent to an enemy. It is still affected by terrain while moving.

Additionally, from the diagram on page 11, we see an example of a Copper Dragon using flying movement. The rules for flying don't explicitly specify that the flying creature can change direction after each square, because that's covered under the normal rules for movement.