打 =手 + 丁. It originally meant “hammer a nail into a plank”, and its meaning was extended widely. In modern Chinese, 打 has many different collocations such as 打理，打包，打听，打扮, 打算, 打游戏，打球，打电话，打车，etc.
Mostly, 打 is used in these two structures:
打 + an object
In this structure, 打 expresses an action or behavior related to this object, e.g. 打游戏 play games; 打车 take a taxi; 打电话 make a call.
汤姆最终养成了成天打游戏的坏习惯。tāng mǔ zuì zhōng yǎng chéng le chéng tiān dǎ diàn nǎo yóu xì de huài xí guàn 。Tom eventually formed the bad habit of playing games all day.
Even if you take a taxi, you’ll still miss you train. 就算你打车，你还是会错过那班火车的。jiù suàn nǐ dǎ chē , nǐ hái shì huì cuò guò nà bān huǒ chē de 。
tā hòu lái gào su wǒ shuō , tā shì dú le nǐ de zhuān lán zhī hòu shòu dào qǐ shì cái gěi wǒ dǎ diàn huà de 。He later told me that he had been inspired to call me after reading in your column.
打 + a verb/adjective
In this structure, a meaning more related to the second character is expressed, e.g. 打扮 dress up; 打算 plan; 打通 put through.
tā men yòng zhū bǎo hé pí yī dǎ bàn qǐ lái , zhǔn bèi chū mén 。They are all dolled up in furs and diamonds and ready to go.
“Do you mean to tell me that you’re going to sanction this marriage?” asked the doctor.
你打算由北边的路去城里吗？ Nǐ dǎsuàn yóu běibian de lù qù chéng lǐ ma? Do you intend to go to the town by the northern route?
她的电话已经坏了，所以我无法打通。tā de diàn huà yǐ jīng huài le , suǒ yǐ wǒ wú fǎ dǎ tōng 。Her telephone had broken down so that I couldn’t get through.
Look up 打 in: