Here's how you solve a problem like this.
You are right that 3 can't go evenly into 8 -- that would be too easy!! :-) If 3 went into 8, all you would need to do is change the 2/3 fraction into 8ths.
Since neither denominator fits evenly into the other, you need to change both fractions. The first step is to find the smallest number that they both go into evenly -- this number is called the LCM or Lowest Common Multiple. To find this, it's usually easiest to just go up the times table for the larger number til you find one that the smaller number also goes into. So, 8 x 1 = 8 (no help); 8 x 2 = 16 (no help); 8 x 3 = 24 (bingo! 3 also goes evenly into 24).
Now you know that you need to change both fractions into 24ths. To do this, multiply both top and bottom by the number needed to get to 24.
1 x 3 3
8 x 3 = 24
2 x 8 16
3 x 8 = 24
So, now you have two fractions you can add together:
3/24 + 16/24 = 19/24
You should always simplify your answer if you can, by dividing top and bottom by the same number. In this case you can't simplify since 19 is a prime number and so can't be divided evenly by any number other than itself and 1.
So, your final answer is 19/24