@Applejack007
Death Note as noted above is a nice starting point. Though it’s less mystery and more cat=and-mouse/battle of wits. Full Metal Alchemist Brotherhood is a good choice too, though once you finish that go check out the 03 FMA. The first ended up in a different direction half way though due to lack of source material at the time but it’s still a good watch regardless, including the movies.
If you want something of a pure mystery you got Detective Conan.
More Drama than mystery, then Gankutsuou
More Action than mystery, then Cowboy Bebop
If you want that along with some yuri under
OVERtones, then watch Puella Magi Madoka Magica.
This should start you on your anime cocktail taste test.