Direction complements are one of the big obstacles in Chinese grammar. The most common ones must be 来(lái) and 去(qù).
Let's have a look at the following combinations:
Up and down: 上 (shàng) and 下 (xià)
For this situation, the up and down directions are covered by 上 and 下.
The structure for this is:
verb [上/下] [来/去]
So you can make any combination you like out of 上 or 下, then 来 or 去. There are four possible combinations there. Let's have a look at an example:
Please come up.
This sounds like it is being said by someone who is upstairs to someone who is downstairs. The speaker wants the listener to make a movement up and towards them, so they use 上 and 来. Notice how English says "come up", whereas Chinese switches the order: 上来 is literally "up come".
这听起来像是楼上的人对楼下的人说的话。说话人想让听话人朝前者的方向向上移动，因此使用了"上"和"来"。注意，英文里说"come up"，而中文则转变顺序变成"上来"，字面意义是"up come"。
What if the person downstairs doesn't want to go up, and asks the first person to come down? They could say:
You come down.
This action is still coming towards the speaker, but now it's down. So 下 and 来 are used.
Let's have a look at some more examples:
cóng bāshí lóu kěyǐ kàn dào zhěnggè chéngshì, zánmen shàngqù kàn yīxià ba.
From the 80th floor you can see the whole city – let's go up and have a look.
zhège fángzi de dìxiàshì tài kǒngbùle, wǒ bù gǎn xiàqù.
The basement in this house is too scary, I don't dare go down.
In and out: 进 (jìn) and 出 (chū)
Besides 上 and 下, you can use a variety of other directions with 来 and 去. One pair is 进 and 出: inward movements and outward movements.
进 and 出 combine with 来 and 去 in the same way as 上 and 下 did above. For example:
Please come in.
The speaker might be in a room or office and is inviting the listener to come in. More examples:
wǒ zài huàn yīfú, nǐ bié jìnlái!
I'm getting changed, don't come in!
wǒ jīn wǎn yào chūqù.
I'm going to go out tonight.
Tā bǎ zìjǐ suǒ zài fángjiān lǐ, bù ràng rén jìnqù!
He's locked himself in his room and won't let anyone in!
With direction complements involving 进 and 出, you've got to consider if the movement is going into or out of a location, and whether the speaker is inside or outside of that location.
Back: 回 (huí)
The next direction we'll look at is 回. This is for movements that are going back or returning. You can combine 回 with 来 or 去 to talk about coming back or going back.
接下来的方向谈到的是"回"。这是用于返回或回归的移动。你可以用"回"搭配"来"或者"去"，可以讲述"coming back (回来)"或"going back (回去)"。
Have a look at some examples:
wǒ wàngle dài sǎn, wǒ yào huíqù ná.
I've forgotten my umbrella. I'll go back and get it.
nǐ shénme shíhou cóng bāxī huílái?
When are you coming back from Brazil?
nǐ jǐ diǎn huíqù?
What time are you going back?
tā xīngqí'èr wǎnshàng huílái.
He's coming back on Tuesday evening.
Across: 过 (guò)
By combining 来 or 去 with 过, you can talk about movements that come across or go across. This can also cover come over and go over in English.
用"过"搭配"来"或者"去"，你可以讲述"come across (过来)"或"go across (过去)"的移动，这也包含英语里"come over"和"go over"的意思。
tā kàn dàole wǒ, jiù guòlái gēn wǒ dǎzhāohū.
He saw me and came over to say hello.
línjū mǎile xīn de jū, wǒ yào guòqù kàn yīxià.
The neighbours have bought a new car – I'm going to go over and have a look.
nǐ yào bùyào guòlái chīfàn?
Do you want to come round to eat?
wǒ kàn dàole yīgè lǎorén shuāidǎo le, mǎshàng jiù guòqù bāngmáng.
I saw an old person fall down, and immediately went over to help.
Note that other than meaning "go over", 过去 is also a word meaning "the past".
注意，"过去"既有"go over"的意思，还是一个表示"the past"的词。
Rising: 起 (qǐ)
In general it means "to rise", and in direction complements it can only be combined with 来. Examples:
qǐng zhàn qǐlái.
Please stand up.
yǐjīng xiàwǔ yīdiǎnle, nǐ kuài diǎn qǐlái!
It's already 1pm – get up now!
Reaching and arriving: 到 (dào)
The structure is slightly different, because you put the destination in between 到 and 来 or 到 and 去.
… 到 [destination] 去
… 到 [destination] 来
These are used to talk about going to places or coming to places. Examples:
wǒ hěn xiǎngdào Běijīng qù.
I'd really like to go to Beijing.
wǒ xīwàng nǐ kěyǐ dào zhèlǐ lái.
I hope you can come here.