The trick is to basically punctuate the whole sentence, not just the bit in quotes. So…
“It was a disaster,” said Dave. “I still can’t believe it.”
It was a disaster said Dave is one sentence, so don’t close the quotes with a full stop. You can use question marks or exclamation points if required.
I still can’t believe it is a full sentence so ends with a stop. (it doesn’t have a dialogue tag such as said Dave after it.
As for the use of single or double quotes, there are two schools of thought:
“And that’s when John said ‘Hi’ to me,” said Dave.
‘And that’s when John said “Hi” to me,’ said Dave.
The first version is norm in America. The second is the norm in the UK. However, as with much these days the lines are blurring. Pick one and make sure you use it consistently.
Never have dialogue from more than one person in the same paragraph, and generally speaking start a new paragraph for each instance of dialogue. But, don’t be frightened to mix a bit of action with the speaking:
“The look on your face is priceless,” said Dave as he toyed with the axe.
John backed off slowly. “There’s no need to get all swingy with that thing.”
The other thing to note is that if a character goes into a monolgue that extends into more than one paragraph then you don’t close the first paragraph with a dialogue mark, but you do start the second one with one:
Dave let the axe fall to the floor. The blood from the blade was going to leave a nasty stain on the carpet but there was nothing he could do about that now. "I thought we were close, John, but you were a bad friend. You messed with the wiring on my kettle causing the explosion that killed my twin brother and you had to pay for that. Finding you was easy, I followed you home from your night class in t-shirt design and lay in wait in your back garden for you to need air. You were bound to need air eventually in this weather, especially with the guilt you’ve been carrying around.
“Still, it’s over now. Live with that. Oh, you can’t. Ha!”
If stuck, flicking through a nearby novel will usually give you the answer.