Translation:Don't give up! Come on!
加油 literally means "add oil". And it is used for cheering in Chinese.
This word cannot be found in classical Chinese and appears only in writings from the last century. The etymology of this word is very controversial. I've seen the following explanations on the Internet:
Refueling vehicles or other machines. As diesel / gasoline engine became popular in 20th century.
Refueling oil lamps.
Adding (squeezing) oil into the container. From chants of oil manufacturers.
Stepping on car pedal (from Wiktionary)
A derivative of Sanskrit verb 'jayati' (win). Similar expression exist in many languages in SE Asia. But this must be a modern borrowing, as 加 is not read as "jia" in Middle Chinese.
Number four is the only one that seems particularly likely to me.
The current description (2018) from Wiktionary:
It started to be used as an interjection in the '60s and '70s during the Macau Grand Prix when people start chanting "加油", urging the drivers to step on the gas pedal to increase speed.
Etymologically it seems to me that it's akin to the German "Gas geben" and the English "step on it".
It has a similar to meaning of "gambatteyo" in Japanese. As Warren pointed out, it means literally to "add oil", and is used when cheering people on during sports events. I think appropriate translations in English would be: come on, hang in there, and keep going.
For me, your 'keep going' is much more natural than 'press on' (hint & accepted) or 'come on' (shown here currently) seems too vague. Similar possibilities might be: 'keep at it,' 'keep trying.' A little farther afield from this that try to capture more the idea of adding fuel: 'work (even) harder' or 'don't let up.'
Depends on the context. I agree with your suggestions, though for cheering at a sports event, I think "come on" and "let's go" work fine. For encouraging a friend, I think "keep at it" is good, and "you can do it" is within the realm of possibility. Also "give 'er", but that's probably regional. :-)
You're definitely right about the choice of an English equivalent depending on the context. Sports did occur to me also, but most other examples in the skill were talking about language learning. You can also go a little colloquial with the idea of ADDING fuel, not simply maintaining effort or not slacking off with something like: 'step (/crank) it up a notch.' But I think that's probably too colloquial or perhaps too regional.
Today marks the day on which "add oil", i.e. the literal translation of the word 加油, gets added into the Oxford English Dictionary.
The direct translation, "Do not give up! Refuel!", should work just as well in this particular case.
Probably, though it wouldn't quite get at the more likely idiomatic meaning (which I think is closer to "put more effort into it", by metaphorical extension from "step on it"), but does nonetheless sound like the beginnings of a good candy bar commercial slogan. :-)