New here, but not new to the stresses of being reserved and shy and trying to flirt with someone I like.
Basically, I turn into a jelly and want to clam up when I try to flirt, but I've found, like other people have suggested, if you ask about the other person and react to things they are saying with more than mmhmms and ahhhs it works out pretty well. You don't need to make innuendo jokes or throw yourself on them, paying attention with good eye contact usually does the trick.
Also smiling doesn't hurt and when you become more comfortable absolutely brush their arm or something to that effect.
Just don't lie or fake interest in something, being genuinely you will attract not only someone you like, but someone who actually likes who you are.
Hope that was useful