Easiest way, a smile and a wink, if they smile back. Initate conversation, if they give you a "wtf" look, it's best to just let that go. Unless you want to try then make yourself known, if you know they are there all the time (say a place they work) then show up and purchase the same thing.
One way I hooked up with a girl who worked at Atlanta Bread Company, I went in every monday, ordered the same thing, with the same change. Cubano Sandwich, with extra sauce on it. She was there every monday. Took me 3 months, mind you, but I got her name after the frist month, her number, after the second, and her.. well.. yea at the end of the third.
That is just one way. But if it's just a chance encounter, you have to show/recieve the signs that your advances are going to work. It's not the movies, a girl who gives you the "eww" face, isn't into you, don't try. The girl who puts out eaisly, is going to cheat on you, don't try (unless you want mushu only) the quiet girl, will take FOREVER to put out, but it's usually worth it. I've never had the quiet girl flip out and pull a whip and handcuffs out while calling her twin sister with gas powered play toys all of a sudden.
"But What do I talk about"
Simple answer, play 21 questions, while throwing in your own experince in. If you don't have any, it's not bad to make up white lies. "Oh, I used to drink and party, but I quit, it's a lon story for a different date" not only does it set you up if you're not a drinker, but using the word 'date' can put fourth that you either a: want to take her out. or. b: are asking her out.
"What signs should I look for?"
This ones is tough. But the standard signs are look or smiles and laughing, even at thinks you don't think are funny, because that means she/he is trying to connect with you. You do the same, if it has a twinge of humor to it, and you see they want a chuckle, laugh, it's alright they just told the worse knock knock joke ever. Infact end the laugh with "that was horrible" and then laugh some more.
Check her stance, if she rigid, then she's defensive, if she's standing loose, swaying some then she's let her defence down (attack now, kidding) also means you're not making her uncomfortable. Ladies here's the thing, guys body language is hard to determine. Our eyes wander, it's natural, we usually stand with our arms crossed, or hand in pockets, and we're normally laid back as it is. But watch the smiles, note the twinges in the coners of the lips, the eye brows raise in a "come hither" perk (see: the hell is that?) we flirt with our faces (that almost got mistyped, can you guess word?) just as you do with your bodies.
"What should I avoid?" (for asking a woman out)
Grim subjects, off race humor, sexism. A woman will come and dispute this, but 90% of women don't want to hear that your great great grand pappy helped start the KKK back when it was just four guys trying to be ghosts to scare the neighborhood kids. Don't want to hear that your cat was ran over by a lawn mower after it od'd on cat nip. Don't want to hear how your kid brother keeps putting your sisters dolls in the kitchen because 'thast where they belong'
These are jokes best set for the dreaded 'friend zone' which is harder than hell to get out of.
"What should I avoid?" (for asking a guy out)
Ladies, don't make yourself seem prude, but don't just throw yoruself out there and seem like a loose goose (see: worst comparison ever) But also don't seem desperate. Remember you have what we want, and any woman can flaunt it, but you must do so with mystery so it leaves us wanting more. Also, we guys, though most won't say anything, don't like a woman dropping 'f-bombs' all the time, especially if we're not cussers ourselves.
Case in point: I currently have a woman who is madly in love with me, but she throws herself in such a distasteful way that I can't find myself really attracted to her, she's beauitful and fun to be with, but she is so distasteful, foul mouthed and it drives me crazy (see: not sexually either)
Be sexy, be confident, be everything you know you are!
"How do I get his/her number?"
Easiest way I do, is 'forget to ask for his/her number after the conversation ends. When you start to head your seperate ways. Count to 5 and then go back to him/her, not in a run, or sprint, but a trot (see: not looking desperate) and say "oh, how am I supposed to ask you out without your number?" THE FACEBOOK FIND/ASK IS HORRIBLE DO NO DO IT!!
"What happens when I ask her/him out"
Think of yourself as a salesman, you're out to prove the product you're pushing is what she's looking for. Be confident, be easy but direct. When he/she says 'yes' don't reply with 'REALLY?!' (see: desperate cry) because that puts out a sign that you've not gone out in a long time, or that you've never gone out ever.
"What if he/she says 'no'?"
Do no.. and I repeat do not take it personally, don't respond with "why not?" or "fine *le sigh*" (see: pity date that ends in heartache for you, and a funny story for him/her to tell their friends) But if they do, just laugh and say 'thats cool' don't hang up, but ask them how their life has been since you last spoke/met. If it's on facebook, don't disconnect, it'll seem like you only wanted to get on to ask that. WHICH I THINK IS REDICULOUS
"Where do I go for a first date?" (for taking a woman out)
This is a totally hard thing to do. Dinner and move? Classic! but you lose the oppertunity to talk to her for 1.5-2 hours and that time to connect cannot be given, because you only have dinner to talk and sell yourself. First, find out what kind of person they are. Don't always have to ask them out before knowing them. And plan accordingly. If she's a fun loving girl, take her to the local rock wall, or go-kart place, then take her to a fun place to eat. For fancy girls, most do like plays and such, and you can do commentaries on the show if it gets boring. Then a nice dinner at a better place (See: Olive Garden isn't fancy)
"Where do I go for a first date?" (for taking a guy out)
His ass should be taking you out, if you're paying for anything. (After asking at least once, and him saying 'no, I got this') then you either 1 totally picked up a dead beat. Or 2 are desperate.
The rest is for you to figure out, make your own way and I know you'll be happier than ever. Not all relationships work out, don't kick yourself if it doesn't. People move, cheat, lie, steal, die, and just move on. You're gonna have to learn to do that last one also