Thank you guys for all the answers.
@zoomkat - Yes, I'll have power.
I personally think that using an ethernet shield is better because i can use the ethernet library which is very nice and easy to understand.
I don't like connecting arduino to the serial port of a router because basically it doesn't have a "direct connection" to the internet. (correct me if I am wrong) This means that there must be sth that constantly controlls the data from the router to the arduino and from the arduino to the router. (much complicated) Although it is the cheapest method I don't really like to try it because I am a beginner with arduino
I am planning to collect the data from the arduino web server page with a VPS with a php cron job and store it in a MySQL database, so that I can play with it in web or software apps.
Final question: Will my arduino have internet access if I put TP-Link TL-WR702N in bridge mode with my home wi-fi network, and simply connect it with the ethernet shield with a lan cable?
I have some other questions but they are not related to networking so I'll open another thread.