Starting conversations with people you don't really know can be very challenging. They could be the "cool" kid on campus, or you might fear looking boring or weird. It is really much easier than you think to start a conversation with someone; most people love meeting and talking to people. The first thing you have to do is be confident in yourself. Just be calm. When starting a conversation you can ask them a question; like if you are in math class, how to solve a problem or something you can both relate to. Then, you can go from there. People love talking about themselves, so let them do most of the talking. Start on some common ground, whether it be a teacher you like/don't like, or the football playoffs, or whatever you think may interest them. Ask them about themselves. In turn most people will ask you about yourself, alas you are in a conversation. Don't ramble on about yourself, tell what is necessary and then get back to them. If they look interested or ask you questions feel free to go into details. Now you should have a conversation going. Allow the conversation to flow in its natural direction, you have broken the awkward stage. If you haven't, just talk more about what they like. Be aware not to overwhelm them. If they appear disinterested or are only offering short, one word answers, just let them be, try again later. Don't be afraid to be yourself when talking to people.
Trying to get into groups is harder, but definitely doable. When trying to get into a group, start off by just hanging around them. It will probably start off by you just kinda standing around listening to them joking and talking since there isn't a history between you and them. Don't just stand near them, stand with them. If they are in a circle, be in the circle. Laugh with them, and give your input when it feels comfortable. Even if you don't talk much, you will be noticed and they will start to associate you with them. Start off by having one on one conversations with different people and getting to know them. This will be a lot easier to do then to just jump into the group and start talking. Even if you just know one of them well, you will start to get to know the others. You may feel a group is clicky and doesn't want to have other members in it, but they probably just seem that way. If you want to be with them, they are not going to just kick you out. They probably just seem clicky because they know each other well and always hang out. My "group" in High School was a very close group. We all knew each other very well, but loved it when we met new people. We thought we were a very open group, but were told that we were clicky even though that was not our intention. Most groups on the outside look like they are fine and don't want new members, but in reality would like to meet new people.
In my experience, I have never met someone who when I tried to talk to them, totally shut me out. Even random people on a bus are willing to strike up a conversation. I go to UCSD so I usually start off a conversation by just making eye contact and asking them "what's up", or "how are you doing". They usually respond with a one word answer, then I introduce myself and start talking about what their major is, or where on campus they live, stuff like that. (I hate talking about school, but it is just something that we all have in common) Once they tell me their major and stuff, i kinda just go from there asking if they like it, or what they want to do with it, and then just see where the conversation goes. Then, next time I see them, I know kind of who they are and can move into talking about other things.