1
Make her feel special. When you see the girl, say hi, make eye contact, give her a wave, and start talking to her. If it's your first time holding hands, then you may both be a little nervous, so it's important to put both of yourselves at ease first. Though holding hands is an innocent gesture, it does come with a level of intimacy,[1] so you should have a good sense that the girl likes you before you go make the move. Even putting your arm around her or putting a hand on her knee can be seen as less intimate than actually holding hands, so try initiating other innocent physical contact before you hold her hand first.

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2
Get closer. If you're sitting, just place your hand down about an inch away from hers. This can help you get a read for how ready she is to hold hands -- if she moves closer to your hand, then she's probably ready for more. If you're standing and walking together, try to get closer to her until your hands are just inches apart.

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3
Make contact. You should make some physical contact before you go ahead and grab her hand. If you're standing, put an arm around her shoulder and brush against her arm until you reach her hand, or keep walking and gently "accidentally" touching hands until taking her hand in yours feels natural. When you're ready to make contact, you can choose from a variety of ways to hold her hand.
Try to be as relaxed as possible before you make contact. The more nervous you are, the more likely you'll be to have sweaty hands! Of course, this is not the worst thing in the world, but if you can avoid it, that's great.
Be as direct as you can instead of fumbling for her hand.[2]
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4
Don't be upset if she's not into it. Contrary to what you may think, not all girls love holding hands. If she pulls away because she's not interested in you, then you'll get the message pretty fast because she'll pull away with her entire body and will look generally uncomfortable. But there's a good chance that she's just not into holding hands because she thinks it's corny or because she's nervous that she has sweaty hands or something, so don't worry too much; you'll figure it out eventually.
If she doesn't want to hold hands, try making a joke to lighten the mood. This way, things won't feel as weird afterward.[3]