Your problem is very common. In spite of the images promoted on television advertisements of suave men that women swarm over, this is not really the experience of most men. While some people are shy and others are more extroverted and gregarious, there are many people who feel uncomfortable and confused when trying to make connections with women they might want to befriend or date. This is certainly true of women as well.
Bars and nightclubs are some of the least ideal places to try and strike up new connections. Fortunately, there are lots of alternatives. While ridiculed in some circles, online dating services are actually a pretty good option for meeting new women. Initiating contact through written correspondence can be considerably less intimidating for people who feel shy or awkward. When a connection progresses to the decision to meet one another, you already have some idea of what the person is like and how you can communicate with one another. If you go this route, just keep your expectations modest . Know that you are likely to get a lot of non-responses as well as a few people who respond. It is not personal, so please don’t interpret it that way. People fail to respond for a huge variety of reasons that have nothing to do with you.
Another option is to pursue activities with groups who share similar interests. This allows connections to develop in a more natural way. Becoming friends with people in a group and noticing who you click with and who notices you gives you time and a degree of comfort that may allow you to approach a new friend or dating partner with more confidence.
Other ways to meet new people include coffee shops, introductions through mutual friends, taking your dog to the dog park, working out at the gym–almost anything you do with regularity that allows for freedom to make casual conversation are good options for forming new connections.
Regarding the issues you raised about awkwardness and how to begin conversations with new people, there are multiple strategies. Joining a mixed gender discussion group, attending a communication workshop, or even joining a group like Toastmasters will give you opportunities to develop these skills and a degree of confidence.
With respect to specific subjects and approaches to starting conversations, begin with the easiest communication of all—a genuine smile. If you are standing near someone you want to talk to you can always talk about the weather (such a cliché but still useful), things happening in the immediate environment (“I really like this park, especially in the summer”), or small compliments directed to the person you are wanting to connect with (be careful not to go overboard or be too personal). In time, you will develop your own approach and rhythm, and the process will gradually become less awkward. And then just watch the women swarm!