Q:

Do not cross the line!

Elliot and Terry used to be friends, until the fateful day when they played a round of that simple, unassuming game. Let’s just say that afterward there were no hugs and kisses to be found between the two...

Across 
Give out punishment
  Protagonists of many YA novels
  Harbor protector  
Down 
Name that fades out of existence?
  What atoms desire
  Sugar or red item

What game did they play? Who went first? And who eventually won the match, thus ruining their friendship?

A:

I think the one who went first was

 Elliot

and the winner was

 also Elliot.

They played

 Tick-tack-toe with "E" and "T". (This is clued by "hugs and kisses", which refers to the use of "xoxo" in letters.)

Answers

 a1. Give out punishment = METE
 a2. Protagonists of many YA novels = TEEN
 a3. Harbor protector = JETTY
 d1. Name that fades out of existence? = PETER
 d2. What atoms desire = OCTETS
 d3. Sugar or red item = BEET

Finished grid:

 

