please treat me well; I look forward to working with you; please remember me
Used when meeting someone for the first time, starting a new relationship, or asking for future cooperation. Conveys a polite request for goodwill.
Nice to meet you, I'm Tanaka. I look forward to working with you.
please do; please take care of
Used when making a request or entrusting something to someone, often with a specific task implied. Softer than a direct command.
I'll be in your care starting tomorrow. Please treat me well.
Please submit these documents by tomorrow. I'm counting on you.
Please take care of receiving the package.
よろしく is the casual, shortened version used among friends, whereas よろしくお願いします is the polite full form suitable for formal or first-meeting situations.
どうぞよろしく is a shorter, more casual greeting used mainly when first meeting someone or sending regards, while よろしくお願いします is more formal and versatile, also used to ask for favors or express ongoing goodwill.
よろしくお願いいたします uses the humble いたします, making it more formal than よろしくお願いします, which uses the plain polite します.
Standard polite phrase; the kanji 願い is normally used.
Ateji spelling using 宜しく; less common but still seen in formal writing.
A set phrase combining よろしく (adverbial form of よろしい, 'good') and お願いします (polite request form of 願う, 'to wish'). The phrase literally means 'please treat me favorably' and has become a standard greeting in Japanese.