The main thing to remember is: you
don't have to tell us their entire life story in the character's opening scenes.
It's really tricky to get character and world development into your scenes in just dialogue and character thoughts without it coming across as telling and infodumps. You want to imply everything about their character with barely a taste of the topic in dialogue. Referencing the topics in passing is more effective than going into deep detail about it, whether it be in narrative or dialogue.
It also depends on the scene. You're not going to be able to build much character in an action scene - although you could show some world and psychological elements like how the character reacts to the action. But if the scene falls flat, then the dialogue will too, so make the scene something rich and interesting with a lot of life and flavour to it.
It's the whole "show, don't tell" rule, and the "less is more" rule, and I find it works best when the characters are involved in some debate or argument, or teasing each other. Quick banter and bickering does a lot more for character development than a wall of text dialogue talking about their life history. Light-hearted banter is a good way to develop supporting characters, I find.
Personality comes from conflict.
The other thing you can do is finish the book then rewrite the first chapters using what you learnt about the characters by the end of the book. That's what I did, and it seemed to work quite well.
In my first book, the first chapter had the protagonist's sister following her around being annoying. It gave me the chance for the sisters to act like sisters and bite at each other's words, giving them plenty of chances to mention different things about each other and the world while still entertaining the reader through humour. Most of the dialogue was quick, snappy statements that dropped hints of bigger issues behind the scenes, or what boy they liked, or the death of their mother, or some of the culture of the world. It was all there in the scene, mentioned in passing, never dropping into anything that I would consider infodumping, and the character development seemed to work quite well. If you want, send me a PM and I could email you the chapter to have a read. It might help.
EDIT: In regards to the 'he said' 'she said' issue. If the dialogue is on the same line as an action that particular character performs, then you don't need the dialogue tag. If the character's are the only two talking, you don't even need the dialogue tags at all after the first two of three bits of dialogue because the reader will be in the flow of switching between heads when they get to each new line. After that, the only time you need to say, he said, again is when a new character joins the conversation.
Once there are three or more characters involved, you either have to use a dialogue tag on just about every line, or you could continue with the action tag trick - which is usually better than saying, he said, on every bit of dialogue.
I pulled this out of the previously mentioned chapter of mine which demonstrates switching between speakers well enough, I think - it even adds a new speaker halfway. Only twice do I use a dialogue tag. The first time to introduce the third character, and the second time just because I thought the action sounded better attached to the dialogue tag than alone.
...
His thumb flicked between the folds, to open the letter again, but Alyce threw herself against him, her fingers digging into his shirt.
Alyce’s pleading face looked up at him. ‘Help me, Papa! Emmi put a lemon peel down my back and I can’t get it out.’
He laughed again. ‘Well that was very cruel of your sister.’
‘I can feel it wriggling around like a worm. Get it out, please.’
‘Why don’t you go get changed? You look like you’ve just become the main ingredient in one of Mrs Potternberry’s plum tarts.’
‘I like Mrs Potternberry’s tarts.’ Alyce’s eyes grew wide. ‘But I don’t want to be in one of them!’
‘Best run along then.’ He patted her back, sending Alyce, flour coated dress, and lemon peel running from the room. He watched her go, scaling the stairs two at a time before vanishing down the hall, then turned back to Emylynn.
‘Is something wrong?’ She asked, with a knowing look in her green eyes, and stepped close.
‘I’ve been summoned to Fort Hastye. I’ll have to leave immediately.’
‘Is it bad?’
‘I don’t know,’ he said, and shrugged.
A sad look crossed her face. ‘What about the pie? Alyce will be upset.’
‘I have no choice.’
...
