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's particularly likely, and it's not controversial so much as a topic that seems to inspire unrigorous folk etymologies. But insofar as "油" is used to denote any number of kinds of oil or related fuel (whether cooking oil, petroleum, gasoline, or any other hydrocarbon), any number of unremarkable folk etymologies can be made up, and any number of related meanings can be played with.
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's akin in meaning to the German "Gas geben" and the English "step on it" ("step on the gas pedal").
Number three is just silly, and the invocation of Sanskrit in number five is extremely far-fetched, given that a "gas station" is a "加油站" ("add-fuel station").
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.
While I'm sure you're joking, and no doubt there are more crowns for you to obtain over time, I recommend DuChinese. There's an app and a web version. I like the latter, even on a mobile device, because it's easier to scroll and to switch between simplified and traditional characters, if you're interested. You can use the free version, but personally I think it's worth the price of subscription, so you can read older articles, multi-part stories, etc.
That's not quite what the article says, though it's unclear. There's an OED blog post ("entry") about "add oil" that seems to suggest it's being considered for inclusion. But it hasn't actually been included in a list of new terms, apparently.
(The problem with this tongue-in-cheek literalism is that "add gas" – or "add petrol", for the British – is a better direct translation, given the implied meaning of "step on it". And a "加油站" isn't an "oil station".)