This is really about interpersonal skills, I believe; how to avoid hurting someone's feelings when they are left out of a situation. You can't. It's human nature to feel this to some extent.
You might possibly soften the blow, however, by:
including them as your semantic group of friends:
another friend and I...; one of my other friends and I...
decreasing their semantic importance:
someone I know and I...; an acquaintance and I...
identifying as belonging to a specific group that doesn't emotionally exclude your friend:
a coworker and I...; a classmate and I...
Not mention the other person at all (this is my preference):
- I went skydiving this weekend. It was terrifying!
- I saw a good movie last weekend.
If the other person asks, Who with?, I will give names, not my association with them. If the other person then asks, Who are they?, I will answer. (a friend from the co-op; a doctor at work; whatever) Usually it doesn't go that far.
There are ways to approach this which involve common sense more than language. There's rarely a reason to start off with A friend and I.... It's not important who you did something with; you just want to share your experience, don't you? Other times, you might want to let your friend you met another interesting person. Just be aware of how you would want to hear it.