Fear of rejection can be absolutely petrifying. I've been there. But you'd be surprised how often you'll feel more relieved than anything when you're given a straightforward answer. It's a whole lot less emotionally taxing than dealing with women who beat around the bush with no real interest in setting a date.

The biggest thing to keep in mind is to not let your whole world start revolving around someone, especially when you've just met. Your relationship with them should not start to become a main priority in your life until at the very least you've entered a serious committed relationship. This will usually come up after you've been on boyfriend/girlfriend terms for a while. You have to first build trust and rapport with someone before that point. Otherwise, you're leaving yourself open to being hurt and taken advantage of.

Likewise, I wouldn't recommend your first move being to open and close with getting her number or asking her on a date. Talk to them first. When you find yourself in a small talk situation, take it. It'll do more than help you gauge her interest. Introduce yourself. Girls can be super into you, but just smiling and staring back doesn't move things forward. She needs to know that you actually talk. Even just talking about the weather or asking if she's seen a movie you've seen is infinitely better than hoping she'll come talk to you first. I'm no expert at timing dates, and I prefer to set them up face-to-face rather than over text, but if she's giving you really good vibes, you can often get her number by the end of your first interaction with her.

Be in the moment. Seriously, don't try and plan a conversation out. You'll just throw yourself off-balance. You can't focus on what she's saying at the same time that you're in your head trying to figure out how to shoehorn-in whatever you planned on saying next. Your subconscious is a hundred times better at sorting this out than you are, so just be in the moment. Thinking of one or two topics you could talk about beforehand is fine, just don't dwell on how you're going to fit them into your interaction.

Eye-contact, eye-contact, eye-contact. I'm not telling you to stare into her soul the whole time you're talking with her, but make an effort to look her directly in the eyes at least when you're striking up your initial conversation/small-talk/whatever, and as you're telling her it was great meeting her and/or asking for her number. Not just when she's talking, but when you're the one talking. Trying to make direct eye-contact is overwhelming for a lot of people, but it goes a long way. You can practice this on women when you're checking out at the grocery store as you walk up to them, and as you're exchanging your have-a-nice-days and thanks-you-toos.

While you're at it, practice exiting a conversation with people. A lot of people can relate to having an otherwise comfortable small-talk taking a turn into awkward silence, knowing that it's an appropriate time for the conversation to end and go your separate ways, but not actually knowing how to end the conversation. There are a lot of dynamics at play depending on things like who approached who. It's easy enough to tell them "thanks, it was nice talking to you" and going on your way when there's a lull in the conversation. However, there are times when the conversation can keep going, but you'd like to continue it another time. A simple "Hey, I've gotta head back" before telling them it was nice talking with them is a tactful way to do this. Practice it with both men and women. It's a skill, and I think not knowing how to do that ends up making it harder to want to approach people to begin with.

I cannot overstate how important it is to get down the basic foundations of talking with people that you have no prior relationship with. We interact with people day-to-day, but if 99% of your experience with other people is with them initiating, carrying, and closing the conversation, it's going to be difficult to do these yourself. Practice these things with the intention to simply build experience and rapport with the people you meet. Many people find that they find partners when they're just socializing without expecting anything to happen.

Build a social circle. It takes time and patience, but put yourself out there, and have at least a couple of different things you spend your time on (exercise, hobbies, etc). These will provide a foundation that will make your romantic pursuits easier and will help you avoid getting into that position where all you have going on in your life is this girl that you just met.

Lastly, and most importantly, be respectful. I don't care if she turned you down or keeps flaking on you. I hear so many personal accounts from women I know that I'm tempted to write another post on the topic. By showing her that you aren't going to blow up on her when things don't go your way, you'll automatically stand out from everyone else. Follow the three-text-rule and don't end up on /r/niceguys. Just move on.