Starting a conversation can definitely be difficult, especially if you don't know the person. I'm fortunate to never really have had any problems with this so I may be able to help.
Making new friends and strengthening connections can give a great feeling. You mention meeting people for the first time so I'll focus on that. With people you don't know, talking about the current situation you're in is always a good idea. If you're meeting new classmates, ask what they did before this. Always introduce yourself either right away or shortly after (say your name, shake hands if it's not a hassle) and make sure you get their name. Try to show interest in what the other person is doing or going to do, and if you're both going to do something you can just talk about that. Making a joke or just a witty comment immediately gives a good first impression.
I could say you should try to keep eye contact, since this will make you appear more confident and open, but some people just aren't comfortable with that. Trying to force yourself to keep looking at someone will only make things worse. Instead try to at least look close to them (you don't have to look them straight in the eyes, maybe slightly above or below). Also, I've seen people stare straight into someone's eyes without a break which can actually be more discomforting. Relax, imitate the other person a little and you should be fine.
Stuttering can be a pain for the person who experiences it but what you should know is that quite literally almost no one will think any less of you because of it. The only thing that matters is how you react to it. And here's how you react to it: you don't. I had a classmate back in high school who had a permanent and instantly noticeable stutter when talking in groups larger than a few people. He didn't look nervous or anything, he just had that stutter. He was very popular and no one ever mentioned it (and that was in high school). Making fun of someone for something they have zero control over is so low that not even the most childish high schoolers will find it clever.
As for what to talk about, you could ask how they're doing out of courtesy but that's not really a conversation topic. They'll probably say they're fine and ask you the same thing. The key in these first conversations is getting to know them better. If there's not much time to chat, try some small talk (difficult, I know) and make sure to give a "talk to you later" or something when you both go elsewhere. Now you can continue the conversation later. If you both have time you can start talking about any subject that seems suitable and continue from there. Be yourself but show interest, ask questions, joke and laugh and tell your own stories (just don't talk endlessly) while trying to keep track of their reactions.
Finally, try to empathize with the person. Some people like talking, some people don't. Some people like listening, some people don't. You'll quickly find that there are people who expect you to lead the conversation and people who almost seem to refuse to let you lead it. Don't freak out if there's nothing left to say. Try to steer the conversation somewhere else or casually leave the conversation if you don't want to continue. It's not your responsibility to entertain them. You don't have to like or be friends with or even talk to every person, but every person you do interact with is another connection made. All these things get easier with experience.
Starting a conversation can definitely be difficult, especially if you don't know the person. I'm fortunate to never really have had any problems with this so I may be able to help.
Making new friends and strengthening connections can give a great feeling. You mention meeting people for the first time so I'll focus on that. With people you don't know, talking about the current situation you're in is always a good idea. If you're meeting new classmates, ask what they did before this. Always introduce yourself either right away or shortly after (say your name, shake hands if it's not a hassle) and make sure you get their name. Try to show interest in what the other person is doing or going to do, and if you're both going to do something you can just talk about that. Making a joke or just a witty comment immediately gives a good first impression.
I could say you should try to keep eye contact, since this will make you appear more confident and open, but some people just aren't comfortable with that. Trying to force yourself to keep looking at someone will only make things worse. Instead try to at least look close to them (you don't have to look them straight in the eyes, maybe slightly above or below). Also, I've seen people stare straight into someone's eyes without a break which can actually be more discomforting. Relax, imitate the other person a little and you should be fine.
Stuttering can be a pain for the person who experiences it but what you should know is that quite literally almost no one will think any less of you because of it. The only thing that matters is how you react to it. And here's how you react to it: you don't. I had a classmate back in high school who had a permanent and instantly noticeable stutter when talking in groups larger than a few people. He didn't look nervous or anything, he just had that stutter. He was very popular and no one ever mentioned it (and that was in high school). Making fun of someone for something they have zero control over is so low that not even the most childish high schoolers will find it clever.
As for what to talk about, you could ask how they're doing out of courtesy but that's not really a conversation topic. They'll probably say they're fine and ask you the same thing. The key in these first conversations is getting to know them better. If there's not much time to chat, try some small talk (difficult, I know) and make sure to give a "talk to you later" or something when you both go elsewhere. Now you can continue the conversation later. If you both have time you can start talking about any subject that seems suitable and continue from there. Be yourself but show interest, ask questions, joke and laugh and tell your own stories (just don't talk endlessly) while trying to keep track of their reactions.
Finally, try to empathize with the person. Some people like talking, some people don't. Some people like listening, some people don't. You'll quickly find that there are people who expect you to lead the conversation and people who almost seem to refuse to let you lead it. Don't freak out if there's nothing left to say. Try to steer the conversation somewhere else or casually leave the conversation if you don't want to continue. It's not your responsibility to entertain them. You don't have to like or be friends with or even talk to every person, but every person you do interact with is another connection made. All these things get easier with experience.