You should start with something that only has a twelve episode run to get a feel for what you like. I think Fooly Cooly only had eight or so episodes, plus, it's wackily funny and shows the weirdness that is Japan.
There are a bajillion different types of anime. Fairytail is good for simple action and weekly battles. Magi and One Piece are along the same vein. I'll say this: Fairtail and One Piece are on, like, episode five hundred each while the second season of Magi is about halfway done at episode thirty-ish. If you can't marathon stuff, the former two will be hard to handle.
If I remember who you are, and I probably don't, you liked Harvest Moon, so I say go harem rom-com. Boku Wa is about the best for that, but maybe not for beginners. I fucking love it, but I tried to get a friend to watch and he was bored.
If you can handle subtitled anime, and you should if you wish to enjoy the best ones that never get dubbed, two very funny animes ended just a week or two ago. Outbreak Company stars a nerdy Japanese NEET who's brought to Middle Earth to teach them about anime. It also has the most moe-tastic girl in the history of aww, Myucel. Funny stuff. Yuusha ni Narenakatta Ore wa Shibushibu Shuushoku wo Ketsui Shimashita is about a hero who has to get a real job when lord Voldermort is slain. The next day Voldermort's daughter walks in and gets a job at the same Target he works at. Boobs jiggle like electric pudding this this one.
Space Dandy starts next week on Adult Swim, in English. It's actually premireing in North America without having to languish in Japan for three years. That would be a pretty good start, It's by the people who did Fullmetal and other legendary anime that you've never heard of before.
It's about a Dandy, in the space.
And yes, the 'the' is supposed to be there cuz' it's a fuckin' boss.