“I was having a conversation with Sarah, and this guy came up and said 'Hi' to her.”

You went to a party and were talking to an acquaintance (Sarah). Someone that she knew said hello and started talking to her. Now it's after the party, and you say this while telling your boyfriend about the situation.

I was having a conversation with Sarah, and this guy came up and said 'Hi' to her.

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I was (doing something), and (clause).

When you're telling a story, you start by giving the setup for the story. This is a description of what was happening before the interesting action started. A very quick way to describe this is to say "I was (doing something), and (something happened). For example, these sentences could be used at the beginning of a short story:

I was washing the dishes, and I noticed that one of the plates had a crack in it.

I was watching TV on the couch, and my wife turned to me and said, "Guess what I heard today..."

The word "and" in these examples means "and then".

have a conversation with (someone)

To "have a conversation with someone" means to talk with someone. If you "talk to" someone, you might just ask a few questions or make small talk. But when you "have a conversation with" someone, it means that you talk for a long time about something interesting.

Here are some different ways to use "conversation". You can use it by itself:

We had a good conversation at the party.

Or you can say who you talked with:

I had a good conversation with Sarah at the party.

You can also say what you talked about:

I had a good conversation with Karen at the party about why she's moving to Boston.

this guy

In the example above, the speaker says "this guy" instead of "a guy". There's a small difference between the two. When you say "this guy" it sounds like you're talking about a specific guy. "A guy" is more general. The identity of the person isn't important.

come up to (someone)

When someone "comes up to" you, it means that they come close to you that they can talk to you. For example:

After my lecture, a student came up to me and told me about a dumb mistake I had made.

You use "come up to" when a person comes to you. But what if you are the one who's speaking? The phrase for this is "go up to":

I went up to the professor after his lecture and pointed out that he had made a mistake on one of the problems.

Another word for this is "approach (someone)". But "approach" is a more formal word.