y=-x2+2x+3=-(x2-2x-3)=-(x-3)(x+1), so the x-intercepts are -1 and 3; the y-intercept is just the constant term 3. x-intercepts are (-1,0) and (3,0). The y-intercept is at (0,3).
To find the vertex:
y=-(x2-2x-3)=-(x2-2x+1-4)=-((x-1)2-4)=-(x-1)2+4. This is called " completing the square".
When x=1, y is maximum at 4, so the vertex is at (1,4).