I can never seem to enjoy animes, I don't know I just cannot take the art style seriously and I don't find it very interesting. The only anime shows I've ever enjoyed were Pokemon and Yu-Gi-Oh I guess.
You just need to find the right one different studios do different art styles you are bound to find one you like. AIC Spirits had by far the best animation of any studio so try them.
Jeez that series was a fucking rollercoaster. One minute it's pure comedy, the next it's uber-drama. It's rare that anything can bring a tear to my eye, least of all anime. Now on to Clannad.
Make sure you don't watch the Aftestory during the weekdays. You will be suffering from depression for one or two days afterwards.
Hetalia,Melancholy of Haruhi Suzamiya,Clannad,Eureka 7,Full Metal Alchemist(and Brotherhood),High School of the dead,Code geasse,Fruits Basket,Moribito Guardian of the spirit,
Higurashi no Naku Koro ni. Actually that's from the first OVA, Rei. The 1st season is dubbed, but it's called When they Cry. The second season is called Higurashi no Naku Koro ni Kai and is subbed. There are two OVAs, Rei and Kira. There's also another series, Umineko no Naku Koro ni, which has some crossover with Higurashi and is similar in premise.