Slayers is the staple of lighthearted fight/magic animes. Like most animes to follow, a lot of it is funny with drama woven in and then takes a serious turn at the end of all three its seasons. The first season of the TV series is a little hard to get through because it’s older and slow paced at times, but it’s worth it overall. The second season called "Slayers Next" is arguably the best season and it’s amazing in all areas of magic, fighting, and humor. The animation is much better. "Slayers Try," the last season, is a nice addition with the best animation of the three. I would recommend staying away from the movies and OVAs (Slayers Glorious, Slayers Excellent, etc.) While decent, the OVAs only feature one character from the original cast.

Visions of Escaflowne –

This is a very good anime with magic and fighting. Most of the fighting is done with a take on the classic mech, but there is definitely a magic element to the whole series. It is heavy on love stories, but there’s plenty of action and dark plot twists. Music is done by Yoko Kanno of Cowboy Bebop, Wolf’s Rain, and Record of Lodoss War fame.

Bastard –

A hilarious and wonderful magic/fight anime. It’s also older, but the animation holds up amazingly well even by today’s standards. The protagonist/anti-hero Dark Schinder is guaranteed to win you over. And the best (and worst) thing is it is very short (only 6 episodes.) Highly recommended

Vampire Hunter D – Bloodlust

I’m not sure how you feel about vampire animes, but this one is gorgeous and the fight scenes are flawless. There is a lot of magic, though perhaps not in the classic anime fighting sense (ie. turn based, video game like fighting). It’s a movie.

Revolutionary Girl Utena –

It definitely has a lot of interpersonal social drama, but there is a lot of fighting and magic elements. You might want to ask a few others that know your taste well if they think you would like this anime. It’s very good, but not for everyone.

Witch Hunter Robin –

Personally I didn’t much care for this series, but you might find it interesting. There is fighting and magic for the most part, but the plot doesn’t go anywhere special.
References :