Since Detective Comics is going to have a new team with many characters who've never met, I figured it would be good to discuss where all these characters stand in regard to their personal relationships with each other (and where they'll likely be at the start of the run): positive, negative, or neutral.
After thinking about it, here's what I've got:
Basil-Bruce: Negative. No reason to think they're on good terms.
Basil-Cass: Neutral. Never met.
Basil-Kate: On the whole, neutral. They worked together briefly and Kate seemed to like him, but if he has amnesia now, who knows?
Basil-Steph: Same as Bruce.
Basil-Tim: Same as Bruce.
Bruce-Cass: Positive. Bruce has taken Cass under his wing.
Bruce-Kate: Positive. They last interacted on good terms, so unless something's happened off-panel, they should still be okay.
Bruce-Steph: Positive. No reason to think they're on bad terms.
Bruce-Tim: Same as Steph.
Cass-Kate: Neutral. Never met.
Cass-Steph: Neutral-to-postive. Steph initially thought Cass was dangerous and referred to her as less than a person due to her lack of speech, but seemed to warm up to her due to Harper's influence.
Cass-Tim: Negative-ish. Tim was kind of a dick to her. Not intentionally, and likely just a result of bad writing, but still.
Kate-Steph: Neutral-ish. Kate personally contacted her during Endgame to give her an assignment, so she knows she exists and doesn't seem to have a problem with her, but nothing else.
Kate-Tim: Similar to Steph. No interactions beyond a brief meeting during Endgame, but didn't seem to have problems.
Tim-Steph: Mostly positive. They've met a few times, but just on "business". There's a hint of something more between them, but not a lot yet.*
* Edited based on jules's input, as seen below.