Cheerful animes like Fruits Basket?

One good resource that I always check for recommendations after finishing a series that I enjoyed is My Anime List.

Definitely check out Ouran High School Host Club, if you haven’t already. It is, in many ways, better than Fruits Basket (characters, pacing, humour), but is still similar enough that it may be a show that you will enjoy.

For something more mature, and bittersweet, I would look to Honey and Clover (school setting, love triangles, identity crises) and Natsume Yuujin Cho (combatting loneliness, building a transnormative family, mythology.)

Also, not in direct relation to your question about shows, but are you feeling depressed? While it is healthy to have a stimulating outlet to unwind to, it may be worth exploring why have been feeling sad and brainstorm other ways that you may be able to cope with those feelings.

I checked out Ouran Host Club. When I read reviews I thought it was innapropriate but it does me as much good as fruits basket. Yep, I will be doing another 3 round on a show just like fruits basket!

Sometimes bittersweet but generally cheerful is Azumanga Daioh. I’d be a little surprised if you haven’t heard of it already, but I very much enjoyed it.

My daughter is wild about Hetalia at the moment. It seems to be pretty light-hearted from what I’ve seen of it. I don’t know Fruits Basket so I couldn’t say how similar it is or not.

Not sure about what you mean by “cheerful”, but these are all quite light-hearted/funny.
Watched them all, and should all be OK to watch. I will put an X in front of those that I would definitely recommend.

X Lucky Star
X Chi’s Sweet Home,
X Bakemonogatari,
Baka to Test to Shoukanjuu,
Bokusatsu Tenshi Dokuro-chan,
Powerpuff Girls Z,
X Midori no Hibi,
FLCL,
Genshiken,
Girls Bravo,
X Great Teacher Onizuka,
Shinryaku Ika Musume,
Hayate no Gotoku,
X Jungle Wa Itsumo Hale Nochi Guu,
X Kyouran Kazoku Nikki,
Makai Senki Disgaea,
Nichijou,
X Ore no Imouto ga Konnani Kawaii Wake ga Nai,
X Panty and Stocking with Garterbelt,
Ranma ½,
X Renkin San-kyuu Magical Pokaan,
Rizelmine,
Hanamaru Youchien,
X Toradora,
Watamote,
X Sayonara Zetsubou Sense

Oh, and you should (in theory, have not tested it) be able to watch all of them here:
http://www.anime-sub.com/

if not, there are always torrents :P