"Let's call a spade a spade" means you want to talk honestly, be blunt, straightforward.
After performing a song at a karaoke bar, I rejoin my friend at our table: "I think I sounded pretty good, what did you think?"
"Let's call a spade a spade - you sounded awful. You couldn't carry a tune if it was strapped to your back."
"Let bygones be bygones" means you want to start fresh with someone with whom you've had a previous disagreement; you want to pretend the argument never happened.
"We've been friends too long to let this little problem come between us. Why don't we just let bygones be bygones, and forget about who owes whom the $5?"