The Olympics

Finally the Olympics started 2 days ago.

While this is grammatically correct, finally is generally used as a transition signal to indicate the end of a sequence of (things, events, etc). Maybe something more explicit like, "After four years of waiting" would fit better--

Everywhere in London looks like a festivals, and everyone looks really happy.

Actually I hadn't liked the London Olympics because it would have been crowded and there were lots of tourists.

This would only make sense if you were writing this after the Olympics was over. It kind of comes across as saying "I actually didn't enjoy watching the "London Olympics". Try something like this instead: "I did not anticipate that everyone would be so happy, since I thought it would be very crowded with all of the tourists around".

And Korea got the gold medal in women's archery too.

Ya it's amazing!

The Olympics finally started 2 days ago.

Adverb placement in English is rather difficult. If you put "finally" at the beginning of a sentence, it does what Andy said. If you put it just before the verb, it means what you wanted it to mean. :)

All of London looks festive and the people look really happy.

Actually, I didn't like the idea of Olympics coming to London because it would be overrun by tourists.

However, I have changed my mind because the Olympic games are really exciting.

I watched the Korean Olympic football match today.

The Korean team had to win because last match was a draw, so it was really important.

"should" is advice, "had to" is necessity.

I cheered while drinking beer. It was so exciting, and we won in the game!

And Korea got the gold medal in women's archery, too.

I am going to watch another football match live in the stadium on Wednesday.

I am expecting we will win!