I'm looking for a news site for Mathematics which particularly covers recently solved mathematical problems together with the unsolved ones. Is there a good site MO users can suggest me or is my only bet just to google for them?
As a counter-point to my somewhat flippant previous answer (which only really applies if one is a specialist in the field), if you are looking at a field in which you are not as much a specialist in, I suggest reading the articles from the Bulletin of the AMS. The articles are designed to be fairly up-to-date and expository in nature, and often gives the state of the art in their reviews.
Of course, a similar caveat as that to Helge's answer applies: the "news" maybe several months out of date. But considering the glacial paces at which a lot of mathematical refereeing takes place, I think it is quite okay.
In the spirit of this answer, you may also find Which journals publish expository work? to be useful.
Another suggestion Annals: to appear. Also other top journals. If a big problem gets solved, its solution probably gets submitted to a journal of this type, so its to appear lists are what you are looking for. Of course, you only learn about the solution of the problem a few years late (refereeing takes time), but you can be almost certain that the solution is actually correct.
The Wikipedia page List of unsolved problems in mathematics has a specific (and long) sublist for recently solved problems.
There is also a list about group theory open problems here : http://www.grouptheory.info/