"Offer" means "do you want this?" or "how about this?". You are offering the other person a choice to do/have something or not to do/have it.
"A offers tea to B" = A says to B "do you want tea?". B has the choice of accepting the tea or not accepting it.
"A offers to drive B home" = A says to B "Do you want me to drive you home?" B can say "yes please" or "no thank you"
"Suggest" means "I think you should do this". You are offering the other person advice.
"A suggests B drink tea" = A says to B "I think you should drink tea". A is not saying "I will give you tea", but "You should drink tea". B can take this advice or not.
"A suggests B consult a lawyer" - A says to B "you should consult a lawyer". A is not offering to be a lawyer, only telling B that B should consult a lawyer. B can take this advice or leave it.
Have a nice day!