I am currently in a conversation with a person I'd like to get to know better. Without trying to come off as cheesy or too flirty, which questions or phrases could I use to make the conversation longer by making the person more inclined to keep conversing with me?
Here are some conversation starters that can help you get to know someone better without being too cheesy or flirty:

What do you like to do for fun? This is a classic conversation starter that can help you get to know the person's interests and hobbies.

What are you currently reading/watching/listening to? This can help you find common interests and give you something to talk about.

Do you have any upcoming travel plans? This can lead to a discussion about travel experiences, favorite destinations, and travel tips.

What are your thoughts on [current event or trending topic]? This can lead to a discussion on a variety of topics, from politics to pop culture.

What is your favorite thing about your job/hobbies/life in general? This can help you understand what the person values and enjoys, and may give you some insight into their personality.

Remember, the key to making a conversation longer and more engaging is to be an active listener and ask follow-up questions based on what the person is saying. Be genuinely interested in what they have to say and show that you are listening by asking thoughtful questions and sharing your own thoughts and experiences.
Those are a bit cheesy. Can you suggest anything more interesting or fun?
You could try a few of the following:
Tell me about your hobbies or interests. I'm always curious to learn more about what people enjoy doing in their free time.
What's something you're really passionate about? I love hearing people talk about their passions and what inspires them.
Have you traveled anywhere recently? I enjoy hearing about people's travel experiences and any interesting stories they have to share.
What do you like to do when you're not working or studying? I'm always interested in learning about how people like to unwind or relax.
Do you have any favorite books, movies, or TV shows? I'm always looking for recommendations and would love to hear what you enjoy.
Tell me about your background or cultural heritage. I find different cultures fascinating and would love to learn more about yours.
What are your future goals or aspirations? I'm interested in hearing about your plans and what you hope to achieve.
Do you have any pets? I'm a big animal lover, and I enjoy hearing about people's pets and the funny or cute things they do.
What's something you've always wanted to learn or try? I'm curious about people's aspirations and the things they want to explore in life.
What's your favorite way to relax or de-stress? I'm always looking for new ideas, and I'd love to hear how you like to take care of yourself.
So what do you think about that new trend about A.I. and all that stuff, you ll see one day people will be asking the A.I. what they should say to ask each other!

And as i suggested remember, the key to making a conversation longer and more engaging is to be an active listener and ask follow-up questions based on what the person is saying. Be genuinely interested in what they have to say and show that you are listening by asking thoughtful questions and sharing your own thoughts and experiences.