Sometimes the difference is explained in terms of meaning, for example, that savoir means “to know how to do something”, whereas connaître means “to be familiar with someone or something”.
In some cases, either can be used, for example:
Tu connais/Tu sais la nouvelle? – Do you know the news?
Pas que je sache/pas à ma connaissance – Not that I know (of).
Perhaps a good rule of thumb is to distinguish the two verbs by what can come after them: savoir is usually followed by a question or clause, an infinitive, or nothing, whereas connaître is usually followed by a noun.
Savez-vous quand il vient? – Do you know when he’s coming?
Je ne savais quoi faire – I didn’t know what to do.
Il ne sait pas ce qu’il veut – He doesn’t know what he wants.
Unlike in English, where the word ‘that’ can be omitted, que can never be omitted:
Je savais qu’il était malade – I knew that he was ill.
Elle sait lire et écrire – she knows how to read and write. (‘How to’ is not translated).
Il sait parler aux enfants – He knows how to talk to children.
Il faut savoir attendre – You have to learn to be patient.
Je (le) sais – I know.
Qui sait? Who knows?
On ne sait jamais – You never know.
Je te le ferai savoir – I’ll let you know.
Tu connais Luc? – Do you know Luc?
Vous connaissez New York? – Are you familiar with New York? (usually meaning ‘Have you personally been to New York?’)
Apprendre à/Chercher à connaître quelqu’un – To get to know/ to try to get to know someone.
Je l’ai connu enfant – I have known him since he was a child.
Tu le connais mal – You are underestimating/misjudging him.
Elle connaît son métier – She really knows her job.
Elle s’y connaît en antiquités – She knows about antiques.
Il ne connaît pas sa force – He doesn’t know his own strength.
Ils ont connu des temps meilleurs – They’ve known better days.
Je ne le connais ni d’Ève ni d’Adam – I don’t know him from Adam.
Nous connaissons de tristes heures – We are going through sad times.
I hope this helps you to know how to translate “to know”!