Do changes in speed always affect orbit size, and vice versa?

[snorkack]

Looked up tangential and radial but had difficulty interpreting the definitions.

Does tangent mean the new orbit would (barely) touch the old orbit's boundary?

Yes.

And does radial mean in direction that's perpendicular to a curve? (like spokes on a wheel)

Yes.
The third direction is axial.
Note that any finite kick at right angle to the speed increases the speed, as per Pythagoras´ theorem. Leaving the speed unchanged requires a kick at obtuse angle.