What consequences to take from what i read in logfiles?

Solution 1:

1. google
2. Yep that's called internet background noise, hackers trying to get into your machine. It's pretty normal. Anyway what you can do is install a host based intrusion detection system like OSSEC. The benefit is that you this will block attacks so you can sleep better at night.

Solution 2:

You must learn your logs and know them by heart. Even if you don't understand the exact meaning of some particular message. But then, when you suddenly see something out of order you will know that you got some problems. This is obvious to you that if you hear that your HD started to make some strange sounds that you never used to hear then something wrong is going with it even if you do not know why it sounds at all, right? The same with logs.

The consequences of being hacked may be very painful. Just imagine that your VPS will be involved into a complex scheme of hacking a financial institution. You have rented this virtual entity and you have a full control. Thus you are fully responsible.

Life is life...