Meeting new people can be exciting, it can also be quite daunting. However, humans are social creatures, we don't live in isolation, so no matter how introverted you are, you are still going to have to learn how to meet new people because many times a man is really as good as his network of friends and acquaintances.
Go Solo to a Bar
This can be an intimidating thing but it is indeed an unusual way to make new friends. However, please avoid going solo to a bar at night time for safety reasons. So, you can basically just go laid-back to a lively bar and seat between tables rather than in an exclusive area to make it easier for you to be seen, and then try to start a conversation with someone or a group of people at another table close to you (though it might be easier if you speak to someone who is also sitting solo like you are). There is really nothing to be afraid of or to get threatened by, the worst that can happen is that the other person or people will kill the conversation, and then you can freely move on to another person or group of people.
Speak Up at Bus Stops
Another way to meet new people is to start a conversation at a bus stop or anywhere you're clumped or stuck with a person or group of people for a period of time. Many great friendships and even marriages have started from this. Just keep a warm and friendly smile and try to start a conversation on general light topics before progressing to deeper subjects (that is, if the conversation goes really well). You should, of course, refrain from sharing sensitive details of your life in the course of the conversation because the person is, after all, still a stranger but you can still be friendly and open without being careless about your personal security.
Funny right? But yes bribe them with small generosities and acts of kindness. This, of course, doesn't mean that you should lavish the person with gifts left, right and center all the time, but when you initially want to start a conversation you can get something for yourself and offer to get for them too. If you're going over to their office or home, you can come with a simple gift (maybe a bottle of wine). Small generosities like this have a way of disarming people and making them more receptive to your friendliness. Just be sure not to take it to heart if your act of generosity is turned down or shunned, it's really not about you (the person doesn't even know you), it's about the person and whatever suspicions or reservations that he/she might have.
Join a Group
This might seem unusual because many people don't realize that a simple way to meet new people is to join a group or an activity that they like. This is such a great and easy way to meet new people because people tend to bond over shared likes and affections. It is also easy to connect with people when everyone is relaxed and doing what they love or like. A work-out class or aerobics class is a good example of such a group. A writing club is also good, a sports club is another great example, and so on.
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