If you write “I am a solution-and-goal-oriented" person you are trying to create a concept of “solution and goal” linked with oriented, and then to associate your linked concept with person.
You are trying to imply the structure:
[{solution and goal}-oriented] person
This method fails because the hyphens do not show the structure of your concepts clearly. They obscure it.
If you write solution and goal-oriented person you associate solution with person as if it were an adjective, which it is not. At best you imply that you are a solution person and a goal-oriented person. This was not your intention.
Best is to regard the hyphens as hooks attached to the preceding word. When followed by a space, the hook links to a following word when it comes. The following word is the one that is attached unspaced to the last of the hooked precedents. Hence:
solution- and goal-oriented" person.
This allows you to create even longer (although unattractive) examples such as mission-, solution- and goal-oriented.