Make a list of Fibonnaci numbers and mark the even ones. Now $f_0$ is even and $f_1$ is odd so the sequence starts even, odd, odd, even, ... Look for a pattern in the occurrence of even Fibonnaci numbers in the sequence, then prove that your pattern must continue indefinitely in the sequence.

Again look for a pattern in the occurrences of multiples of 3 in the Fibonnaci sequence. To prove the pattern always applies use the Fibonnaci difference relation $f_{n+2}=f_{n+1}+f_n$ repeatedly to show that if a certain term is divisible by 3 then other terms further along the sequence will also be divisible by 3.