Are you playing in casual or ranked? You should probably be able to get to rank 20 in ranked mode fairly easily, as you can't loose ranks until you get there so there's not really any competition (everyone in rank 25 to 21 is going to be ether very new, very casual, or very bad). Once you do get there though, yeah, it can be hard for a new player, as is perhaps the way of games like this it typically just becomes harder and harder over time to be a new player (although standard years helps a bit). This weeks tavern brawl opens in an hour, most tavern brawls are mostly aimed towards more casual random fun and you don't need a collection (although not every brawl is like that). For some stupendously stupid reason you need to be at least level 20 in one of your classes in order to enter the brawl though, which you may or may not have (I'd suggest finding a class you like and focusing on it to get to said level 20... Honestly some weeks I don't play anything but the tavern brawl, but I still do all my quests and such, so it might be worth getting at least that far and seeing if you like playing the brawls.)
For some reason the card game has oceans in it, so my main accounts in NA, although I do have an EU account past the tutorial if you wanta add me. (Criptfeind#1207) I'd pretty much never be on though.
In the process of checking that I did in fact have an EU account I opened a pack I've had sitting in my EU account for like, 8 months or something and got a two legendaries. Whereas in my main account my last 60 packs have had no legendaries at all. Thanks blizzard. Really rubbing it in.
Edit: Incidentally, this weeks tavern brawl is pick three cards and you get 10 of each in your deck.