It is tough to recover from a Google Penalty but the first thing you should do is find out why you got the penalty. For example, whether it is a Panda or Penguin penalty. If it is a Penguin penalty then it is caused by low quality backlinks. In that case, you should either remove the low quality backlinks or use the disavow tool. The disavow tool needs to be used with caution because it can potentially remove the effects of high quality backlinks if used incorrectly. Once you have done that you should build high quality links and wait for Google to re-rank your website.
The Panda penalty is given to low quality website. For example, websites with low quality or spun content. This is a bit more difficult because it is the website itself causing the penalty and the best thing to do is to turn your website in a high quality website with lots of very detailed and unique content.
This is a very good question! Google won't inform you that they gave you a penalty but there are signs to shows that you have one. You may get warnings in your Google Webmaster tools for various reasons like spammy backlinks or an increase in 404 errors. If you have a warning for spammy backlinks then that is a sign of a Penguin penalty. With a penalty, you lose rankings and hence traffic so if you suddenly lost a lot of organic traffic and you don't rank well in specific keywords anymore, then these are signs of a Google penalty.
If you want to check if you have a Google Penalty, I advice you do the following:
Check Google analytics to see if you have a large change in traffic (don't use other tools like AW stats as they are not very accurate)
Look at keywords that you rank well in and see if they are still doing well