From the list, from up to down: Series that I really really recommended for anyone that havent watched CGDCT:
1. K-On S1 + S2
2. Minami-ke (still have guys, and a lot of season)
3. Maidragon (havent watch, but heard it so good, similar style to lucky star)
4. Saki (best if you learn about mahjong rules first)
5. Lyrical Nanoha - Watch 1st and 2nd movie, it kinda standalone (retelling 1st and 2nd season)
6. Mai-Hime - Think battle series like naruto, but in high school (not all girls)
7. Shirobako - Best to learn about anime industry as well
8. Madoka Magica - Get at least Blu-ray (kinda remaster of tv series) or movie version
9. Non-non biyori - Kinda tell you how the lifestyle in rural area
10. Hibike! Euphonium - Kinda like sport anime since everyone aiming for 1 goal
11. Idolmaster - Out of all idol anime, this kinda complete package (make sure you didnt get xenoglossia or cinderella version since both kinda a bit weaker)
12. Nichijou - Watch principal vs deer first on youtube, if you enjoy that, then you can enjoy the series
13. Railgun - Watch Index S1 first at least, other stuff is optional
There are probably other, but Im going with the one that I highly rated and I think it doesnt require specific niche fandom first (like lucky star need you to be knowledge-able in otaku culture)