Hey Elina welcome to Res!
There's lots of fun things to do on here (:
Like playing games and collecting pets, and socialising. A lot of the dailies can be fun and can also help you earn a little bit of tu on the games side of things some examples are Precious Pearl Quandary, Egg count, twuntie cay and Mystery box. (These are in the luck and skill category) Although these don't give you loads of tu they still count. Once you've saved up a little it might be a good idea to invest in stocks this is a great way to earn higher amounts of tu. There's some great stock guides around and if you'd like I can send you one.
You can visit the apple tree each day to get free apples for your pets. Also if you go to Atquati there are a couple places that are worth visiting, first is the enchanted springs, this one gives you free items, stat changes or tu but it can also take them, However once you reach a higher level by going consecutively each day you can even get colour changes for your pets! At the forest you can collect a flower and also explore to pick up any pets that other users have abandoned. And lastly on Relcore there is a trash where you can pick up one item a day. There are many more great things to do on here but those are just a few tips to get you started.
Florence also has a great thread with info here http://www.rescreatu.com/forum/cat/general/introductions/a-guide-for-new-users/~page/1/
Hope this helped <3